Rectangular Coordinates

Transcription

1 Rectangular Coordinates What ou should learn Plot points in the Cartesian plane. Use the Distance Formula to find the distance between two points. Use the Midpoint Formula to find the midpoint of a line segment. Use a coordinate plane and geometric formulas to model and solve real-life problems. Wh ou should learn it The Cartesian plane can be used to represent relationships between two variables. For instance, in Eercise 0, a graph represents the minimum wage in the United States from 90 to 00. The Cartesian Plane Just as ou can represent real numbers b points on a real number line, ou can represent ordered pairs of real numbers b points in a plane called the rectangular coordinate sstem, or the Cartesian plane, named after the French mathematician René Descartes (9 0). The Cartesian plane is formed b using two real number lines intersecting at right angles, as shown in Figure. The horizontal real number line is usuall called the -ais, and the vertical real number line is usuall called the -ais. The point of intersection of these two aes is the origin, and the two aes divide the plane into four parts called quadrants. Quadrant II Origin -ais Quadrant I (Vertical number line) (Horizontal number line) Quadrant III Quadrant IV -ais -ais Directed distance (, ) Directed distance -ais FIGURE FIGURE Simulation.. FIGURE Video (, ) (, ) (0, 0) (, ) (, 0) Each point in the plane corresponds to an ordered pair (, ) of real numbers and, called coordinates of the point. The -coordinate represents the directed distance from the -ais to the point, and the -coordinate represents the directed distance from the -ais to the point, as shown in Figure. Directed distance from -ais, Directed distance from -ais The notation, denotes both a point in the plane and an open interval on the real number line. The contet will tell ou which meaning is intended. Eample Plotting Points in the Cartesian Plane Plot the points (, ), (, ), (0, 0), (, 0), and (, ). To plot the point (, ), imagine a vertical line through on the -ais and a horizontal line through on the -ais. The intersection of these two lines is the point,. The other four points can be plotted in a similar wa, as shown in Figure. Now tr Eercise.

2 The beaut of a rectangular coordinate sstem is that it allows ou to see relationships between two variables. It would be difficult to overestimate the importance of Descartes s introduction of coordinates in the plane. Toda, his ideas are in common use in virtuall ever scientific and business-related field. Eample Sketching a Scatter Plot Year, t Amount, A From 990 through 00, the amounts A (in millions of dollars) spent on skiing equipment in the United States are shown in the table, where t represents the ear. Sketch a scatter plot of the data. (Source: National Sporting Goods Association) To sketch a scatter plot of the data shown in the table, ou simpl represent each pair of values b an ordered pair (t, A) and plot the resulting points, as shown in Figure. For instance, the first pair of values is represented b the ordered pair 990, 7. Note that the break in the t-ais indicates that the numbers between 0 and 990 have been omitted. Amount Spent on Skiing Equipment Dollars (in millions) A t Year FIGURE Now tr Eercise. In Eample, ou could have let t represent the ear 990. In that case, the horizontal ais would not have been broken, and the tick marks would have been labeled through (instead of 990 through 00). Technolog The scatter plot in Eample is onl one wa to represent the data graphicall. You could also represent the data using a bar graph and a line graph. If ou have access to a graphing utilit, tr using it to represent graphicall the data given in Eample.

3 a + b = c The Pthagorean Theorem and the Distance Formula The following famous theorem is used etensivel throughout this course. a c Pthagorean Theorem For a right triangle with hpotenuse of length c and sides of lengths a and b, ou have a b c, as shown in Figure. (The converse is also true. That is, if a b c, then the triangle is a right triangle.) FIGURE.. Video Video b (, ) (, ) d (, ) Suppose ou want to determine the distance d between two points, and, in the plane. With these two points, a right triangle can be formed, as shown in Figure. The length of the vertical side of the triangle is, and the length of the horizontal side is B the Pthagorean Theorem, ou can write d. d. This result is the Distance Formula. The Distance Formula The distance d between the points, and, in the plane is d. FIGURE Video Eample. Finding a Distance Find the distance between the points, and,. Algebraic Let,, and,,. Then appl the Distance Formula. d Distance Formula Substitute for,,, and. Simplif. Simplif..8 Use a calculator. So, the distance between the points is about.8 units. You can use the Pthagorean Theorem to check that the distance is correct. Graphical Use centimeter graph paper to plot the points A, and B,. Carefull sketch the line segment from A to B. Then use a centimeter ruler to measure the length of the segment. 7 cm d? Pthagorean Theorem? Substitute for d. Distance checks. Now tr Eercises (a) and (b). FIGURE 7 The line segment measures about.8 centimeters, as shown in Figure 7. So, the distance between the points is about.8 units.

4 7 (, 7) d = d = 0 d = (, ) (, 0) 7 FIGURE 8 Eample Verifing a Right Triangle Show that the points,,, 0, and, 7 are vertices of a right triangle. The three points are plotted in Figure 8. Using the Distance Formula, ou can find the lengths of the three sides as follows. Because d 7 9 d 0 d d d 0 d ou can conclude b the Pthagorean Theorem that the triangle must be a right triangle. Now tr Eercise. The Midpoint Formula To find the midpoint of the line segment that joins two points in a coordinate plane, ou can simpl find the average values of the respective coordinates of the two endpoints using the Midpoint Formula. The Midpoint Formula The midpoint of the line segment joining the points, and, is given b the Midpoint Formula Midpoint,. Video Eample Finding a Line Segment s Midpoint. (9, ) (, 0) 9 (, ) Midpoint FIGURE 9 Find the midpoint of the line segment joining the points, and 9,. Let,, and, 9,. Midpoint, 9,, 0 Midpoint Formula Substitute for,,, and. Simplif. The midpoint of the line segment is, 0, as shown in Figure 9. Now tr Eercise (c).

5 Simulation Simulation. Simulation. Simulation. Football Pass. Distance (in ards) 0 (0, 8) 0 0 (0, ) Distance (in ards) FIGURE 0 Applications Eample Finding the Length of a Pass During the third quarter of the 00 Sugar Bowl, the quarterback for Louisiana State Universit threw a pass from the 8-ard line, 0 ards from the sideline. The pass was caught b a wide receiver on the -ard line, 0 ards from the same sideline, as shown in Figure 0. How long was the pass? You can find the length of the pass b finding the distance between the points 0, 8 and 0,. d So, the pass was about 0 ards long. Now tr Eercise 7. Distance Formula Substitute for,,, and. Simplif. Simplif. Use a calculator. In Eample, the scale along the goal line does not normall appear on a football field. However, when ou use coordinate geometr to solve real-life problems, ou are free to place the coordinate sstem in an wa that is convenient for the solution of the problem. Eample 7 Estimating Annual Revenue Revenue (in billions of dollars) 0 FIGURE FedE Annual Revenue (00,.7) (00,.) Midpoint (00, 0.) Year FedE Corporation had annual revenues of $0. billion in 00 and $.7 billion in 00. Without knowing an additional information, what would ou estimate the 00 revenue to have been? (Source: FedE Corp.) One solution to the problem is to assume that revenue followed a linear pattern. With this assumption, ou can estimate the 00 revenue b finding the midpoint of the line segment connecting the points 00, 0. and 00,.7. Midpoint, , 00,. Midpoint Formula Substitute for,,, and. Simplif. So, ou would estimate the 00 revenue to have been about $. billion, as shown in Figure. (The actual 00 revenue was $. billion.) Now tr Eercise 9.

6 . Much of computer graphics consists of transformations of points in a coordinate plane. One tpe of transformation, a translation, is illustrated in Eample 8. Other tpes include reflections, rotations, and stretches. Simulation Eample 8 Translating Points in the Plane The triangle in Figure has vertices at the points,,,, and,. Shift the triangle three units to the right and two units upward and find the vertices of the shifted triangle, as shown in Figure. (, ) (, ) 7 7 (, ) FIGURE FIGURE To shift the vertices three units to the right, add to each of the -coordinates. To shift the vertices two units upward, add to each of the -coordinates. Original Point Translated Point,,,,,,,,, Now tr Eercise. The figures provided with Eample 8 were not reall essential to the solution. Nevertheless, it is strongl recommended that ou develop the habit of including sketches with our solutions even if the are not required. The following geometric formulas are used at various times throughout this course. For our convenience, these formulas along with several others are also provided on the inside back cover of this tet. Common Formulas for Area A, Perimeter P, Circumference C, and Volume V Rectangle Circle Triangle Rectangular Solid Circular Clinder Sphere A lw A r A bh V lwh V r h V r P l w C r P a b c w l r a h b c w h l r h r

7 Eample 9 Using a Geometric Formula FIGURE cm h A clindrical can has a volume of 00 cubic centimeters cm and a radius of centimeters (cm), as shown in Figure. Find the height of the can. The formula for the volume of a clinder is V r h. To find the height of the can, solve for h. h V r Then, using V 00 and r, find the height. h Substitute 00 for V and for r. Simplif denominator..98 Use a calculator. Because the value of h was rounded in the solution, a check of the solution will not result in an equalit. If the solution is valid, the epressions on each side of the equal sign will be approimatel equal to each other. V r h Write original equation. 00?.98 Substitute 00 for V, for r, and.98 for h checks. You can also use unit analsis to check that our answer is reasonable. 00 cm cm.98 cm Now tr Eercise. W RITING ABOUT MATHEMATICS Etending the Eample Eample 8 shows how to translate points in a coordinate plane. Write a short paragraph describing how each of the following transformed points is related to the original point. Original Point,,, Transformed Point,,,

8 Eercises The smbol Click on Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarged cop of the graph. to view the Make a Decision eercise. VOCABULARY CHECK. Match each term with its definition. (a) -ais (i) point of intersection of vertical ais and horizontal ais (b) -ais (ii) directed distance from the -ais (c) origin (iii) directed distance from the -ais (d) quadrants (iv) four regions of the coordinate plane (e) -coordinate (v) horizontal real number line (f) -coordinate (vi) vertical real number line In Eercises, fill in the blanks. Glossar. An ordered pair of real numbers can be represented in a plane called the rectangular coordinate sstem or the plane.. The is a result derived from the Pthagorean Theorem.. Finding the average values of the representative coordinates of the two endpoints of a line segment in a coordinate plane is also known as using the. In Eercises and, approimate the coordinates of the points... In Eercises, plot the points in the Cartesian plane..... D B,,,, 0,,, 0, 0,,,,,,, 8, 0.,,,,,.,,,, A C,,, In Eercises 7 0, find the coordinates of the point. 7. The point is located three units to the left of the -ais and four units above the -ais. 8. The point is located eight units below the -ais and four units to the right of the -ais. 9. The point is located five units below the -ais and the coordinates of the point are equal. 0. The point is on the -ais and units to the left of the -ais. D C B A In Eercises 0, determine the quadrant(s) in which (, ) is located so that the condition(s) is (are) satisfied.. > 0 and < 0. < 0 and < 0. and > 0. > and. <. > 7. < 0 and > 0 8. > 0 and < 0 9. > 0 0. < 0 In Eercises and, sketch a scatter plot of the data shown in the table.. Number of Stores The table shows the number of Wal-Mart stores for each ear from 99 through 00. (Source: Wal-Mart Stores, Inc.) Year, Number of stores,

9 . Meteorolog The table shows the lowest temperature on record (in degrees Fahrenheit) in Duluth, Minnesota for each month, where represents Januar. (Source: NOAA) In Eercises, find the distance between the points. (Note: In each case, the two points lie on the same horizontal or vertical line.) In Eercises 7 0, (a) find the length of each side of the right triangle, and (b) show that these lengths satisf the Pthagorean Theorem (, ) (0, ) Month,.,,,.,, 8,.,,,.,,, (, ) (9, ) (9, ) (, ) 8 Temperature, (, 0) 8 (, 0) (, ) (, ) (, ) (, ) In Eercises 0, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points..,, 9, 7.,,, 0., 0,,. 7,,, 8.,,,., 0, 0, 7.,,, 8.,,, 9..,.,.7, ,.,.,.9 In Eercises and, show that the points form the vertices of the indicated polgon.. Right triangle:, 0,,,,. Isosceles triangle:,,,,,. A line segment has, as one endpoint and m, m as its midpoint. Find the other endpoint, of the line segment in terms of,, m, and m.. Use the result of Eercise to find the coordinates of the endpoint of a line segment if the coordinates of the other endpoint and midpoint are, respectivel, (a),,, and (b),,,.. Use the Midpoint Formula three times to find the three points that divide the line segment joining, and, into four parts.. Use the result of Eercise to find the points that divide the line segment joining the given points into four equal parts. (a),,, (b),, 0, 0 7. Sports A soccer plaer passes the ball from a point that is 8 ards from the endline and ards from the sideline. The pass is received b a teammate who is ards from the same endline and 0 ards from the same sideline, as shown in the figure. How long is the pass? Distance (in ards) (, 8) (0, ) Distance (in ards) 8. Fling Distance An airplane flies from Naples, Ital in a straight line to Rome, Ital, which is 0 kilometers north and 0 kilometers west of Naples. How far does the plane fl?

10 Sales In Eercises 9 and 0, use the Midpoint Formula to estimate the sales of Big Lots, Inc. and Dollar Tree Stores, Inc. in 00, given the sales in 00 and 00. Assume that the sales followed a linear pattern. (Source: Big Lots, Inc.; Dollar Tree Stores, Inc.) 9. Big Lots 0. Dollar Tree In Eercises, the polgon is shifted to a new position in the plane. Find the coordinates of the vertices of the polgon in its new position... (, ) (, ) Year units (, ) units Sales (in millions) 00 $ 00 $7 Year Sales (in millions) 00 $ $800 units (, ) 7 (, ) units (, 0) (, ). Original coordinates of vertices: 7,,,,,, 7, Shift: eight units upward, four units to the right. Original coordinates of vertices:, 8,,, 7,,, Shift: units downward, 0 units to the left Retail Price In Eercises and, use the graph below, which shows the average retail price of pound of butter from 99 to 00. (Source: U.S. Bureau of Labor Statistics). Approimate the highest price of a pound of butter shown in the graph. When did this occur?. Approimate the percent change in the price of butter from the price in 99 to the highest price shown in the graph. Advertising In Eercises 7 and 8, use the graph below, which shows the cost of a 0-second television spot (in thousands of dollars) during the Super Bowl from 989 to 00. (Source: USA Toda Research and CNN) Cost of 0-second TV spot (in thousands of dollars) 7. Approimate the percent increase in the cost of a 0-second spot from Super Bowl XXIII in 989 to Super Bowl XXXV in Estimate the percent increase in the cost of a 0-second spot (a) from Super Bowl XXIII in 989 to Super Bowl XXVII in 99 and (b) from Super Bowl XXVII in 99 to Super Bowl XXXVII in Music The graph shows the numbers of recording artists who were elected to the Rock and Roll Hall of Fame from 98 to 00. Number elected Year Average price (in dollars per pound) Year (a) Describe an trends in the data. From these trends, predict the number of artists elected in 008. (b) Wh do ou think the numbers elected in 98 and 987 were greater in other ears? Year

11 0. Labor Force Use the graph below, which shows the minimum wage in the United States (in dollars) from 90 to 00. (Source: U.S. Department of Labor) Minimum wage (in dollars) Model It Year (a) Which decade shows the greatest increase in minimum wage? (b) Approimate the percent increases in the minimum wage from 990 to 99 and from 99 to 00. (c) Use the percent increase from 99 to 00 to predict the minimum wage in 008. (d) Do ou believe that our prediction in part (c) is reasonable? Eplain.. Sales The Coca-Cola Compan had sales of $8, million in 99 and $,900 million in 00. Use the Midpoint Formula to estimate the sales in 998, 000, and 00. Assume that the sales followed a linear pattern. (Source: The Coca-Cola Compan). Data Analsis: Eam Scores The table shows the mathematics entrance test scores and the final eamination scores in an algebra course for a sample of 0 students (a) Sketch a scatter plot of the data. (b) Find the entrance eam score of an student with a final eam score in the 80s. (c) Does a higher entrance eam score impl a higher final eam score? Eplain.. Volume of a Billiard Ball A billiard ball has a volume of.9 cubic inches. Find the radius of a billiard ball.. Length of a Tank The diameter of a clindrical propane gas tank is feet. The total volume of the tank is 0. cubic feet. Find the length of the tank.. Geometr A Slow Moving Vehicle sign has the shape of an equilateral triangle. The sign has a perimeter of 9 centimeters. Find the length of each side of the sign. Find the area of the sign.. Geometr The radius of a traffic cone is centimeters and the lateral surface of the cone is 7 square centimeters. Find the height of the cone. 7. Dimensions of a Room A room is. times as long as it is wide, and its perimeter is meters. (a) Draw a diagram that represents the problem. Identif the length as l and the width as w. (b) Write l in terms of w and write an equation for the perimeter in terms of w. (c) Find the dimensions of the room. 8. Dimensions of a Container The width of a rectangular storage container is. times its height. The length of the container is inches and the volume of the container is 000 cubic inches. (a) Draw a diagram that represents the problem. Label the height, width, and length accordingl. (b) Write w in terms of h and write an equation for the volume in terms of h. (c) Find the dimensions of the container. 9. Data Analsis: Mail The table shows the number of pieces of mail handled (in billions) b the U.S. Postal Service for each ear from 99 through 00. (Source: U.S. Postal Service) Year, Pieces of mail, (a) Sketch a scatter plot of the data. (b) Approimate the ear in which there was the greatest decrease in the number of pieces of mail handled. (c) Wh do ou think the number of pieces of mail handled decreased?

12 70. Data Analsis: Athletics The table shows the numbers of men s M and women s W college basketball teams for each ear from 99 through 00. (Source: National Collegiate Athletic Association) Year, Men s Women s teams, M teams, W (a) Sketch scatter plots of these two sets of data on the same set of coordinate aes. (b) Find the ear in which the numbers of men s and women s teams were nearl equal. (c) Find the ear in which the difference between the numbers of men s and women s teams was the greatest. What was this difference? 7. Make a Conjecture Plot the points,,,, and 7, on a rectangular coordinate sstem. Then change the sign of the -coordinate of each point and plot the three new points on the same rectangular coordinate sstem. Make a conjecture about the location of a point when each of the following occurs. (a) The sign of the -coordinate is changed. (b) The sign of the -coordinate is changed. (c) The signs of both the - and -coordinates are changed. 7. Collinear Points Three or more points are collinear if the all lie on the same line. Use the steps below to determine if the set of points A,, B,, C, and the set of points A 8,, B,, C, are collinear. (a) For each set of points, use the Distance Formula to find the distances from A to B, from B to C, and from A to C. What relationship eists among these distances for each set of points? (b) Plot each set of points in the Cartesian plane. Do all the points of either set appear to lie on the same line? (c) Compare our conclusions from part (a) with the conclusions ou made from the graphs in part (b). Make a general statement about how to use the Distance Formula to determine collinearit. Snthesis True or False? In Eercises 7 and 7, determine whether the statement is true or false. Justif our answer. 7. In order to divide a line segment into equal parts, ou would have to use the Midpoint Formula times. 7. The points 8,,,, and, represent the vertices of an isosceles triangle. 7. Think About It When plotting points on the rectangular coordinate sstem, is it true that the scales on the - and -aes must be the same? Eplain. 7. Proof Prove that the diagonals of the parallelogram in the figure intersect at their midpoints. FIGURE FOR 7 FIGURE FOR In Eercises 77 80, use the plot of the point 0, 0 in the figure. Match the transformation of the point with the correct plot. [The plots are labeled (a), (b), (c), and (d).] (a) (b) (c) (0, 0) (d) 77. 0, , , 0 0, 0 Skills Review ( b, c ) ( a + b, c) ( a, 0) In Eercises 8 88, solve the equation or inequalit < < 88. (, ) 0 0

13 Graphs of Equations What ou should learn Sketch graphs of equations. Find - and -intercepts of graphs of equations. Use smmetr to sketch graphs of equations. Find equations of and sketch graphs of circles. Use graphs of equations in solving real-life problems. Wh ou should learn it The graph of an equation can help ou see relationships between real-life quantities. For eample, in Eercise 7, a graph can be used to estimate the life epectancies of children who are born in the ears 00 and 00. The Graph of an Equation In the previous section, ou used a coordinate sstem to represent graphicall the relationship between two quantities. There, the graphical picture consisted of a collection of points in a coordinate plane. Frequentl, a relationship between two quantities is epressed as an equation in two variables. For instance, 7 is an equation in and. An ordered pair a, b is a solution or solution point of an equation in and if the equation is true when a is substituted for and b is substituted for. For instance,, is a solution of 7 because 7 is a true statement. In this section ou will review some basic procedures for sketching the graph of an equation in two variables. The graph of an equation is the set of all points that are solutions of the equation. Eample Determining s Determine whether (a), and (b), are solutions of the equation 0 7. a. 0 7? 0 7 Write original equation. Substitute for and for., is a solution. Because the substitution does satisf the original equation, ou can conclude that the ordered pair, is a solution of the original equation. b. 0 7? Write original equation. Substitute for and for., is not a solution. Because the substitution does not satisf the original equation, ou can conclude that the ordered pair, is not a solution of the original equation. Now tr Eercise. The basic technique used for sketching the graph of an equation is the point-plotting method. Video. Video. Sketching the Graph of an Equation b Point Plotting. If possible, rewrite the equation so that one of the variables is isolated on one side of the equation.. Make a table of values showing several solution points.. Plot these points on a rectangular coordinate sstem.. Connect the points with a smooth curve or line.

14 Eample Sketching the Graph of an Equation Sketch the graph of 7. Because the equation is alread solved for, construct a table of values that consists of several solution points of the equation. For instance, when, 7 0 which implies that, 0 is a solution point of the graph. 7, 0, , 7,,,, From the table, it follows that, 0, 0, 7,,,,,,, and, are solution points of the equation. After plotting these points, ou can see that the appear to lie on a line, as shown in Figure. The graph of the equation is the line that passes through the si plotted points. (, 0) 8 (0, 7) (, ) (, ) 8 0 (, ) (, ) FIGURE Now tr Eercise.

15 Eample Sketching the Graph of an Equation Sketch the graph of. Because the equation is alread solved for, begin b constructing a table of values. One of our goals in this course is to learn to classif the basic shape of a graph from its equation. For instance, ou will learn that the linear equation in Eample has the form m b and its graph is a line. Similarl, the quadratic equation in Eample has the form a b c and its graph is a parabola. 0 7,,, 0,,,, 7 Net, plot the points given in the table, as shown in Figure. Finall, connect the points with a smooth curve, as shown in Figure 7. (, ) (, ) (, ) (, 7) (, ) (0, ) (, ) (, ) (, ) (, 7) (, ) (0, ) = FIGURE FIGURE 7 Now tr Eercise 7. The point-plotting method demonstrated in Eamples and is eas to use, but it has some shortcomings. With too few solution points, ou can misrepresent the graph of an equation. For instance, if onl the four points,,,,,, and, in Figure were plotted, an one of the three graphs in Figure 8 would be reasonable. FIGURE 8

16 No -intercepts; one -intercept Technolog To graph an equation involving and on a graphing utilit, use the following procedure.. Rewrite the equation so that is isolated on the left side.. Enter the equation into the graphing utilit.. Determine a viewing window that shows all important features of the graph.. Graph the equation. For more etensive instructions on how to use a graphing utilit to graph an equation, see the Graphing Technolog Guide on the tet website at college.hmco.com. Three -intercepts; one -intercept One -intercept; two -intercepts Intercepts of a Graph It is often eas to determine the solution points that have zero as either the -coordinate or the -coordinate. These points are called intercepts because the are the points at which the graph intersects or touches the - or -ais. It is possible for a graph to have no intercepts, one intercept, or several intercepts, as shown in Figure 9. Note that an -intercept can be written as the ordered pair, 0 and a -intercept can be written as the ordered pair 0,. Some tets denote the -intercept as the -coordinate of the point a, 0 [and the -intercept as the -coordinate of the point 0, b ] rather than the point itself. Unless it is necessar to make a distinction, we will use the term intercept to mean either the point or the coordinate. Finding Intercepts. To find -intercepts, let be zero and solve the equation for.. To find -intercepts, let be zero and solve the equation for. Video Video Video Video. No intercepts FIGURE 9 = (, 0) FIGURE 0 (0, 0) (, 0) Eample Finding - and -Intercepts Find the - and -intercepts of the graph of. Let 0. Then 0 has solutions 0 and ±. -intercepts: 0, 0,, 0,, 0 Let 0. Then 0 0 has one solution, 0. -intercept: 0, 0 See Figure 0. Now tr Eercise.

17 .. Video Video Smmetr Graphs of equations can have smmetr with respect to one of the coordinate aes or with respect to the origin. Smmetr with respect to the -ais means that if the Cartesian plane were folded along the -ais, the portion of the graph above the -ais would coincide with the portion below the -ais. Smmetr with respect to the -ais or the origin can be described in a similar manner, as shown in Figure. (, ) (, ) (, ) (, ). Video (, ) (, ) -ais smmetr -ais smmetr Origin smmetr FIGURE Knowing the smmetr of a graph before attempting to sketch it is helpful, because then ou need onl half as man solution points to sketch the graph. There are three basic tpes of smmetr, described as follows. Simulation. 7 (, 7) (, 7) (, ) (, ) (, ) (, ) FIGURE = -ais smmetr Graphical Tests for Smmetr. A graph is smmetric with respect to the -ais if, whenever, is on the graph,, is also on the graph.. A graph is smmetric with respect to the -ais if, whenever, is on the graph,, is also on the graph.. A graph is smmetric with respect to the origin if, whenever, is on the graph,, is also on the graph. Eample Testing for Smmetr The graph of is smmetric with respect to the -ais because the point, is also on the graph of. (See Figure.) The table below confirms that the graph is smmetric with respect to the -ais. 7 7,, 7,,,,, 7 Now tr Eercise.

18 Algebraic Tests for Smmetr. The graph of an equation is smmetric with respect to the -ais if replacing with ields an equivalent equation.. The graph of an equation is smmetric with respect to the -ais if replacing with ields an equivalent equation.. The graph of an equation is smmetric with respect to the origin if replacing with and with ields an equivalent equation. (, 0) FIGURE = (, ) (, ) Notice that when creating the table in Eample, it is easier to choose -values and then find the corresponding -values of the ordered pairs. Eample Using Smmetr as a Sketching Aid Use smmetr to sketch the graph of. Of the three tests for smmetr, the onl one that is satisfied is the test for -ais smmetr because is equivalent to. So, the graph is smmetric with respect to the -ais. Using smmetr, ou onl need to find the solution points above the -ais and then reflect them to obtain the graph, as shown in Figure., 0, 0,, Now tr Eercise 7. Eample 7 Sketching the Graph of an Equation (, ) = (, ) (, ) (, ) (0, ) (, ) FIGURE (, 0) Sketch the graph of. This equation fails all three tests for smmetr and consequentl its graph is not smmetric with respect to either ais or to the origin. The absolute value sign indicates that is alwas nonnegative. Create a table of values and plot the points as shown in Figure. From the table, ou can see that 0 when. So, the -intercept is 0,. Similarl, 0 when. So, the -intercept is, 0. 0, 0,, 0,, 0,,, Now tr Eercise.

19 Throughout this course, ou will learn to recognize several tpes of graphs from their equations. For instance, ou will learn to recognize that the graph of a second-degree equation of the form Center: (h, k) a b c is a parabola (see Eample ). The graph of a circle is also eas to recognize. Radius: r Point on circle: (, ) Circles Consider the circle shown in Figure. A point, is on the circle if and onl if its distance from the center h, k is r. B the Distance Formula, h k r. FIGURE B squaring each side of this equation, ou obtain the standard form of the equation of a circle. To find the correct h and k, from the equation of the circle in Eample 8, it ma be helpful to rewrite the quantities and, using subtraction., So, h and k. Standard Form of the Equation of a Circle The point, lies on the circle of radius r and center (h, k) if and onl if h k r. From this result, ou can see that the standard form of the equation of a circle with its center at the origin, h, k 0, 0, is simpl r. Circle with center at origin Video Eample 8 Finding the Equation of a Circle. The point, lies on a circle whose center is at,, as shown in Figure. Write the standard form of the equation of this circle. The radius of the circle is the distance between, and,. r h k Distance Formula Substitute for,, h, and k. (, ) Simplif. Simplif. (, ) 0 Radius Using h, k, and r 0, the equation of the circle is h k r Equation of circle 0 Substitute for h, k, and r. 0. Standard form FIGURE Now tr Eercise.

20 . Simulation You should develop the habit of using at least two approaches to solve ever problem. This helps build our intuition and helps ou check that our answer is reasonable. Height, Weight, Application In this course, ou will learn that there are man was to approach a problem. Three common approaches are illustrated in Eample 9. A Numerical Approach: Construct and use a table. A Graphical Approach: Draw and use a graph. An Algebraic Approach: Use the rules of algebra. Eample 9 Recommended Weight The median recommended weight (in pounds) for men of medium frame who are to 9 ears old can be approimated b the mathematical model , where is the man s height (in inches). (Source: Metropolitan Life Insurance Compan) a. Construct a table of values that shows the median recommended weights for men with heights of,,, 8, 70, 7, 7, and 7 inches. b. Use the table of values to sketch a graph of the model. Then use the graph to estimate graphicall the median recommended weight for a man whose height is 7 inches. c. Use the model to confirm algebraicall the estimate ou found in part (b). a. You can use a calculator to complete the table, as shown at the left. b. The table of values can be used to sketch the graph of the equation, as shown in Figure 7. From the graph, ou can estimate that a height of 7 inches corresponds to a weight of about pounds. Weight (in pounds) FIGURE 7 7 Recommended Weight Height (in inches) c. To confirm algebraicall the estimate found in part (b), ou can substitute 7 for in the model. 0.07(7).99(7) So, the graphical estimate of pounds is fairl good. Now tr Eercise 7.

21 Eercises The smbol indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. Click on to view the complete solution of the eercise. Click on to print an enlarged cop of the graph. Click on to view the Make a Decision eercise. VOCABULARY CHECK: Fill in the blanks. Glossar. An ordered pair a, b is a of an equation in and if the equation is true when a is substituted for and b is substituted for.. The set of all solution points of an equation is the of the equation.. The points at which a graph intersects or touches an ais are called the of the graph.. A graph is smmetric with respect to the if, whenever, is on the graph,, is also on the graph.. The equation h k r is the standard form of the equation of a with center and radius.. When ou construct and use a table to solve a problem, ou are using a approach. In Eercises, determine whether each point lies on the graph of the equation. Equation Points. (a) 0, (b),. (a), 0 (b), 8. (a), (b), 0. (a), (b), 9 In Eercises 8, complete the table. Use the resulting solution points to sketch the graph of the equation , 0, 0, 8. 0, In Eercises 9 0, find the - and -intercepts of the graph of the equation

22 In Eercises, assume that the graph has the indicated tpe of smmetr. Sketch the complete graph of the equation ais smmetr.. Origin smmetr -ais smmetr -ais smmetr In Eercises, use the algebraic tests to check for smmetr with respect to both aes and the origin In Eercises, use smmetr to sketch the graph of the equation In Eercises, use a graphing utilit to graph the equation. Use a standard setting. Approimate an intercepts The smbol 8 In Eercises 7, write the standard form of the equation of the circle with the given characteristics. 7. Center: 0, 0 ; radius: 8. Center: 0, 0 ; radius: 9. Center:, ; radius: 0. Center: 7, ; radius: 7. Center:, ; solution point: 0, 0. Center:, ; solution point:,. Endpoints of a diameter: 0, 0,, 8. Endpoints of a diameter:,,, In Eercises 70, find the center and radius of the circle, and sketch its graph Depreciation A manufacturing plant purchases a new molding machine for $,000. The depreciated value (reduced value) after t ears is given b,000 0,000t, 0 t 8. Sketch the graph of the equation. 7. Consumerism You purchase a jet ski for $800. The depreciated value after t ears is given b t, 0 t. Sketch the graph of the equation. 7. Geometr A regulation NFL plaing field (including the end zones) of length and width has a perimeter of 00 or ards. (a) Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. (b) Show that the width of the rectangle is 0 and its area is indicates an eercise or a part of an eercise in which ou are instructed to use a graphing utilit. A 0. (c) Use a graphing utilit to graph the area equation. Be sure to adjust our window settings. (d) From the graph in part (c), estimate the dimensions of the rectangle that ield a maimum area. (e) Use our school s librar, the Internet, or some other reference source to find the actual dimensions and area of a regulation NFL plaing field and compare our findings with the results of part (d).

23 7. Geometr A soccer plaing field of length and width has a perimeter of 0 meters. (a) Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. (b) Show that the width of the rectangle is w 80 and its area is A 80. (c) Use a graphing utilit to graph the area equation. Be sure to adjust our window settings. (d) From the graph in part (c), estimate the dimensions of the rectangle that ield a maimum area. (e) Use our school s librar, the Internet, or some other reference source to find the actual dimensions and area of a regulation Major League Soccer field and compare our findings with the results of part(d). Model It 7. Population Statistics The table shows the life epectancies of a child (at birth) in the United States for selected ears from 90 to 000. (Source: U.S. National Center for Health Statistics) 7. Electronics The resistance (in ohms) of 000 feet of solid copper wire at 8 degrees Fahrenheit can be approimated b the model 0, , 00 where is the diameter of the wire in mils (0.00 inch). (Source: American Wire Gage) (a) Complete the table (b) Use the table of values in part (a) to sketch a graph of the model. Then use our graph to estimate the resistance when 8.. (c) Use the model to confirm algebraicall the estimate ou found in part (b). (d) What can ou conclude in general about the relationship between the diameter of the copper wire and the resistance? Year Life epectanc, Snthesis A model for the life epectanc during this period is 0.00t 0.7t., 0 t 00 where represents the life epectanc and t is the time in ears, with t 0 corresponding to 90. (a) Sketch a scatter plot of the data. (b) Graph the model for the data and compare the scatter plot and the graph. (c) Determine the life epectanc in 98 both graphicall and algebraicall. (d) Use the graph of the model to estimate the life epectancies of a child for the ears 00 and 00. (e) Do ou think this model can be used to predict the life epectanc of a child 0 ears from now? Eplain. True or False? In Eercises 77 and 78, determine whether the statement is true or false. Justif our answer. 77. A graph is smmetric with respect to the -ais if, whenever, is on the graph,, is also on the graph. 78. A graph of an equation can have more than one -intercept. 79. Think About It Suppose ou correctl enter an epression for the variable on a graphing utilit. However, no graph appears on the displa when ou graph the equation. Give a possible eplanation and the steps ou could take to remed the problem. Illustrate our eplanation with an eample. 80. Think About It Find a and b if the graph of a b is smmetric with respect to (a) the -ais and (b) the origin. (There are man correct answers.) Skills Review 8. Identif the terms: Rewrite the epression using eponential notation. ( ) In Eercises 8 88, simplif the epression t 88.

24 Linear Equations in Two Variables What ou should learn Use slope to graph linear equations in two variables. Find slopes of lines. Write linear equations in two variables. Use slope to identif parallel and perpendicular lines. Use slope and linear equations in two variables to model and solve real-life problems. Wh ou should learn it Linear equations in two variables can be used to model and solve real-life problems. For instance, in Eercise 09, ou will use a linear equation to model student enrollment at the Pennslvania State Universit. Using Slope The simplest mathematical model for relating two variables is the linear equation in two variables m b. The equation is called linear because its graph is a line. (In mathematics, the term line means straight line.) B letting 0, ou can see that the line crosses the -ais at b, as shown in Figure 8. In other words, the -intercept is 0, b. The steepness or slope of the line is m. m b Slope -Intercept The slope of a nonvertical line is the number of units the line rises (or falls) verticall for each unit of horizontal change from left to right, as shown in Figure 8 and Figure 9. -intercept (0, b) unit = m + b m units, m > 0 (0, b) unit -intercept m units, m < 0 = m + b Positive slope, line rises. Negative slope, line falls. FIGURE 8 FIGURE 9 A linear equation that is written in the form m b written in slope-intercept form. is said to be The Slope-Intercept Form of the Equation of a Line The graph of the equation m b is a line whose slope is m and whose -intercept is 0, b.. Video Eploration Use a graphing utilit to compare the slopes of the lines m, where m 0.,,, and. Which line rises most quickl? Now, let m 0.,,, and. Which line falls most quickl? Use a square setting to obtain a true geometric perspective. What can ou conclude about the slope and the rate at which the line rises or falls?

25 (, ) = (, ) FIGURE 0 Slope is undefined. Once ou have determined the slope and the -intercept of a line, it is a relativel simple matter to sketch its graph. In the net eample, note that none of the lines is vertical. A vertical line has an equation of the form a. Vertical line The equation of a vertical line cannot be written in the form m b because the slope of a vertical line is undefined, as indicated in Figure 0. Eample Graphing a Linear Equation Sketch the graph of each linear equation. a. b. c. a. Because b, the -intercept is 0,. Moreover, because the slope is m, the line rises two units for each unit the line moves to the right, as shown in Figure. b. B writing this equation in the form 0, ou can see that the -intercept is 0, and the slope is zero. A zero slope implies that the line is horizontal that is, it doesn t rise or fall, as shown in Figure. c. B writing this equation in slope-intercept form Write original equation. Subtract from each side. Write in slope-intercept form. ou can see that the -intercept is 0,. Moreover, because the slope is m, the line falls one unit for each unit the line moves to the right, as shown in Figure. = + = = + (0, ) m = (0, ) m = 0 (0, ) m = When m is positive, the line rises. FIGURE When m is 0, the line is horizontal. FIGURE When m is negative, the line falls. FIGURE Now tr Eercise 9.

26 Finding the Slope of a Line (, ) Given an equation of a line, ou can find its slope b writing the equation in slope-intercept form. If ou are not given an equation, ou can still find the slope of a line. For instance, suppose ou want to find the slope of the line passing through the points, and,, as shown in Figure. As ou move from left to right along this line, a change of units in the vertical direction corresponds to a change of units in the horizontal direction. FIGURE (, ) and the change in rise the change in run The ratio of to represents the slope of the line that passes through the points, and,. Slope change in change in rise run Video The Slope of a Line Passing Through Two Points The slope m of the nonvertical line through, and, is. Video m where.. When this formula is used for slope, the order of subtraction is important. Given two points on a line, ou are free to label either one of them as, and the other as,. However, once ou have done this, ou must form the numerator and denominator using the same order of subtraction. m m m Correct Correct Incorrect For instance, the slope of the line passing through the points, and, 7 can be calculated as m 7 or, reversing the subtraction order in both the numerator and denominator, as m 7.

27 Eample Finding the Slope of a Line Through Two Points Find the slope of the line passing through each pair of points. a., 0 and, b., and, c. 0, and, d., and, a. Letting,, 0 and,,, ou obtain a slope of m 0. See Figure. b. The slope of the line passing through, and, is m 0 0. See Figure. c. The slope of the line passing through 0, and, is m 0 See Figure 7. d. The slope of the line passing through, and, is m 0.. See Figure 8. Because division b 0 is undefined, the slope is undefined and the line is vertical. In Figures to 8, note the relationships between slope and the orientation of the line. a. Positive slope: line rises from left to right b. Zero slope: line is horizontal c. Negative slope: line falls from left to right d. Undefined slope: line is vertical (, 0) FIGURE FIGURE (0, ) m = m = (, ) (, ) FIGURE 7 FIGURE 8 Now tr Eercise. (, ) (, ) m = 0 Slope is undefined. (, ) (, )

28 Writing Linear Equations in Two Variables If, is a point on a line of slope m and, is an other point on the line, then m. This equation, involving the variables and, can be rewritten in the form m which is the point-slope form of the equation of a line. Video. Video. Point-Slope Form of the Equation of a Line The equation of the line with slope m passing through the point, is m. The point-slope form is most useful for finding the equation of a line. You should remember this form. FIGURE 9 = (, ) When ou find an equation of the line that passes through two given points, ou onl need to substitute the coordinates of one of the points into the point-slope form. It does not matter which point ou choose because both points will ield the same result. Eample Using the Point-Slope Form Find the slope-intercept form of the equation of the line that has a slope of and passes through the point,. Use the point-slope form with m and,,. m Point-slope form Substitute for m,, and. Simplif. Write in slope-intercept form. The slope-intercept form of the equation of the line is. The graph of this line is shown in Figure 9. Now tr Eercise 9. The point-slope form can be used to find an equation of the line passing through two points, and,. To do this, first find the slope of the line m, and then use the point-slope form to obtain the equation. Two-point form This is sometimes called the two-point form of the equation of a line.

29 ... Eploration Find d and d in terms of m and m, respectivel (see figure). Then use the Pthagorean Theorem to find a relationship between and m. FIGURE 0 m (0, 0) Video Video d d = + (, m ) (, m ) (, ) = = Technolog 7 On a graphing utilit, lines will not appear to have the correct slope unless ou use a viewing window that has a square setting. For instance, tr graphing the lines in Eample using the standard setting 0 0 and 0 0. Then reset the viewing window with the square setting 9 9 and. On which setting do the lines and appear to be perpendicular? Parallel and Perpendicular Lines Slope can be used to decide whether two nonvertical lines in a plane are parallel, perpendicular, or neither. Parallel and Perpendicular Lines. Two distinct nonvertical lines are parallel if and onl if their slopes are equal. That is, m m.. Two nonvertical lines are perpendicular if and onl if their slopes are negative reciprocals of each other. That is, m m. Video Eample Finding Parallel and Perpendicular Lines Find the slope-intercept forms of the equations of the lines that pass through the point, and are (a) parallel to and (b) perpendicular to the line. B writing the equation of the given line in slope-intercept form Write original equation. Subtract from each side. Write in slope-intercept form. ou can see that it has a slope of m, as shown in Figure 0. a. An line parallel to the given line must also have a slope of. So, the line through, that is parallel to the given line has the following equation. Write in point-slope form. Multipl each side b. Distributive Propert 7 Write in slope-intercept form. b. An line perpendicular to the given line must have a slope of because is the negative reciprocal of. So, the line through, that is perpendicular to the given line has the following equation. Write in point-slope form. Multipl each side b. Distributive Propert Write in slope-intercept form. Now tr Eercise 9. Notice in Eample how the slope-intercept form is used to obtain information about the graph of a line, whereas the point-slope form is used to write the equation of a line.

30 Applications In real-life problems, the slope of a line can be interpreted as either a ratio or a rate. If the -ais and -ais have the same unit of measure, then the slope has no units and is a ratio. If the -ais and -ais have different units of measure, then the slope is a rate or rate of change. Eample Using Slope as a Ratio The maimum recommended slope of a wheelchair ramp is A business is installing a wheelchair ramp that rises inches over a horizontal length of feet. Is the ramp steeper than recommended? (Source: Americans with Disabilities Act Handbook) The horizontal length of the ramp is feet or 88 inches, as shown in Figure. So, the slope of the ramp is Because Slope vertical change horizontal change 0.08, in. 88 in the slope of the ramp is not steeper than recommended. in. ft FIGURE Now tr Eercise 97. Eample Using Slope as a Rate of Change Cost (in dollars) 0,000 9,000 8,000 7,000,000,000,000,000,000,000 FIGURE C Manufacturing C= + 00 Marginal cost: m = $ Fied cost: $ Number of units Production cost A kitchen appliance manufacturing compan determines that the total cost in dollars of producing units of a blender is C 00. Cost equation Describe the practical significance of the -intercept and slope of this line. The -intercept 0, 00 tells ou that the cost of producing zero units is $00. This is the fied cost of production it includes costs that must be paid regardless of the number of units produced. The slope of m tells ou that the cost of producing each unit is $, as shown in Figure. Economists call the cost per unit the marginal cost. If the production increases b one unit, then the margin, or etra amount of cost, is $. So, the cost increases at a rate of $ per unit. Now tr Eercise 0.

31 Section. Linear Equations in Two Variables 8 Most business epenses can be deducted in the same ear the occur. One eception is the cost of propert that has a useful life of more than ear. Such costs must be depreciated (decreased in value) over the useful life of the propert. If the same amount is depreciated each ear, the procedure is called linear or straight-line depreciation. The book value is the difference between the original value and the total amount of depreciation accumulated to date. Eample 7 Straight-Line Depreciation. Value (in dollars) FIGURE Video,000 0,000 8,000,000,000,000 Useful Life of Equipment V (0,,000) V= 0 t +,000 (8, 000) 8 0 Number of ears Straight-line depreciation t A college purchased eercise equipment worth $,000 for the new campus fitness center. The equipment has a useful life of 8 ears. The salvage value at the end of 8 ears is $000. Write a linear equation that describes the book value of the equipment each ear. Let V represent the value of the equipment at the end of ear t. You can represent the initial value of the equipment b the data point 0,,000 and the salvage value of the equipment b the data point 8, 000. The slope of the line is 000,000 m $0 8 0 which represents the annual depreciation in dollars per ear. Using the pointslope form, ou can write the equation of the line as follows. V,000 0 t 0 Write in point-slope form. V 0t,000 Write in slope-intercept form. The table shows the book value at the end of each ear, and the graph of the equation is shown in Figure. Year, t Value, V 0,000 0,70 9,00 8,0 7,000,70,00 7,0 8,000 Now tr Eercise 07. In man real-life applications, the two data points that determine the line are often given in a disguised form. Note how the data points are described in Eample 7.

32 Eample 8 Predicting Sales per Share The sales per share for Starbucks Corporation were $.97 in 00 and $8.7 in 00. Using onl this information, write a linear equation that gives the sales per share in terms of the ear. Then predict the sales per share for 00. (Source: Starbucks Corporation) Sales per share (in dollars) FIGURE Starbucks Corporation (, 9.97) (, 8.7) (,.97) =.t +.7 Year ( 00) t Let t represent 00. Then the two given values are represented b the data points,.97 and, 8.7. The slope of the line through these points is m.. Using the point-slope form, ou can find the equation that relates the sales per share and the ear t to be.97. t.t.7. Write in point-slope form. Write in slope-intercept form. According to this equation, the sales per share in 00 was..7 $9.97, as shown in Figure. (In this case, the prediction is quite good the actual sales per share in 00 was $0..) Now tr Eercise 09. Given points Estimated point Linear etrapolation FIGURE Given points Linear interpolation FIGURE Estimated point The prediction method illustrated in Eample 8 is called linear etrapolation. Note in Figure that an etrapolated point does not lie between the given points. When the estimated point lies between two given points, as shown in Figure, the procedure is called linear interpolation. Because the slope of a vertical line is not defined, its equation cannot be written in slope-intercept form. However, ever line has an equation that can be written in the general form A B C 0 General form where A and B are not both zero. For instance, the vertical line given b a can be represented b the general form a 0. Summar of Equations of Lines. General form: A B C 0. Vertical line:. Horizontal line:. Slope-intercept form:. Point-slope form: a b m b m. Two-point form:

33 Eercises The smbol Click on Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarged cop of the graph. to view the Make a Decision eercise. VOCABULARY CHECK: Fill in the blanks. In Eercises, fill in the blanks. Glossar. The simplest mathematical model for relating two variables is the equation in two variables m b.. For a line, the ratio of the change in to the change in is called the of the line.. Two lines are if and onl if their slopes are equal.. Two lines are if and onl if their slopes are negative reciprocals of each other.. When the -ais and -ais have different units of measure, the slope can be interpreted as a.. The prediction method is the method used to estimate a point on a line that does not lie between the given points. 7. Match each equation of a line with its form. (a) A B C 0 (i) Vertical line (b) a (ii) Slope-intercept form (c) b (iii) General form (d) m b (iv) Point-slope form (e) m (v) Horizontal line In Eercises and, identif the line that has each slope.. (a) m. (a) m 0 (b) m is undefined. (b) m (c) m (c) m In Eercises and, sketch the lines through the point with the indicated slopes on the same set of coordinate aes. Point Slopes., (a) 0 (b) (c) (d)., (a) (b) (c) (d) Undefined In Eercises 8, estimate the slope of the line... 8 L L L 8 8 L 8 L L In Eercises 9 0, find the slope and -intercept (if possible) of the equation of the line. Sketch the line In Eercises 8, plot the points and find the slope of the line passing through the pair of points..,,,.,,,.,,,. 0, 0,, 0.,,,. 7 8,,, 7..8,.,.,. 8..7, 8.,.,.

34 In Eercises 9 8, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are man correct answers.) Point Slope 9. 0.,, m 0 m is undefined...., 0, 8, m m m is undefined ,, 0, 9 7,, m 0 m m m m In Eercises 9 0, find the slope-intercept form of the equation of the line that passes through the given point and has the indicated slope. Sketch the line. Point Slope , 0, 0, 0, 0, 0,, m m m m m m m is undefined.. 0, m is undefined ,,.,.8., 8. m 0 m 0 m m In Eercises, find the slope-intercept form of the equation of the line passing through the points. Sketch the line..,,,. (, ), (, ). 8,, 8, 7.,,,.,,,.,,, 7. 0,, 9 0, 9 8.,,, , 0.,, 0. 8, 0.,,.,,,,,, 7, 8, 7,.,,., 0. In Eercises 8, determine whether the lines L and L passing through the pairs of points are parallel, perpendicular, or neither.. L : 0,,, 9. L :,,, L : 0,,, 7. L :,,, 0 8. L : (, 8), (, ) L : 0,,, 7 L :,,, In Eercises 9 78, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. Point 9., 70., 7., , 7., 0 7., 7., 7., 77.., ,. Line In Eercises 79 8, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts a, 0 and 0, b is a, b 79. -intercept:, intercept:, 0 -intercept: 0, -intercept: 0, 8. -intercept:, intercept:, 0 -intercept: 0, -intercept: 0, 8. Point on line:, 8. Point on line:, -intercept: c, 0 -intercept: d, 0 -intercept: 0, c, c 0 -intercept: 0, d, d 0 Graphical Interpretation In Eercises 8 88, identif an relationships that eist among the lines, and then use a graphing utilit to graph the three equations in the same viewing window. Adjust the viewing window so that the slope appears visuall correct that is, so that parallel lines appear parallel and perpendicular lines appear to intersect at right angles. 8. (a) (b) (c) 8. (a) (b) (c) a 0, b L :,,, 9

35 87. (a) (b) (c) 88. (a) 8 (b) (c) In Eercises 89 9, find a relationship between and such that, is equidistant (the same distance) from the two points. 89.,,, 90.,,, 8 9.,, 7, 9.,, 7, 9. Sales The following are the slopes of lines representing annual sales in terms of time in ears. Use the slopes to interpret an change in annual sales for a one-ear increase in time. (a) The line has a slope of m. (b) The line has a slope of m 0. (c) The line has a slope of m Revenue The following are the slopes of lines representing dail revenues in terms of time in das. Use the slopes to interpret an change in dail revenues for a one-da increase in time. (a) The line has a slope of m 00. (b) The line has a slope of m 00. (c) The line has a slope of m Average Salar The graph shows the average salaries for senior high school principals from 990 through 00. (Source: Educational Research Service) Salar (in dollars) 8,000 80,000 7,000 70,000,000 0,000,000 (, 9,77) (, 8,9) (0, 79,89) (8, 7,80) (,,99) (,,78) (0,,7) 8 0 Year (0 990) (a) Use the slopes to determine the time periods in which the average salar increased the greatest and the least. (b) Find the slope of the line segment connecting the ears 990 and 00. (c) Interpret the meaning of the slope in part (b) in the contet of the problem. 9. Net Profit The graph shows the net profits (in millions) for Applebee s International, Inc. for the ears 99 through 00. (Source: Applebee s International, Inc.) Net profit (in millions of dollars) (8, 0.7) (9, 7.) (, 8.0) (7,.) (, 9.) (,.) (0,.) (, 99.) (, 8.0) (, 8.) Year ( 99) (a) Use the slopes to determine the ears in which the net profit showed the greatest increase and the least increase. (b) Find the slope of the line segment connecting the ears 99 and 00. (c) Interpret the meaning of the slope in part (b) in the contet of the problem. 97. Road Grade You are driving on a road that has a % uphill grade (see figure). This means that the slope of the road is 00. Approimate the amount of vertical change in our position if ou drive 00 feet. 98. Road Grade From the top of a mountain road, a surveor takes several horizontal measurements and several vertical measurements, as shown in the table ( and are measured in feet) (a) Sketch a scatter plot of the data. (b) Use a straightedge to sketch the line that ou think best fits the data. (c) Find an equation for the line ou sketched in part (b). (d) Interpret the meaning of the slope of the line in part (c) in the contet of the problem. (e) The surveor needs to put up a road sign that indicates the steepness of the road. For instance, a surveor would put up a sign that states 8% grade on a road with a downhill grade that has a slope of What should the sign state for the road in this problem?

36 Rate of Change In Eercises 99 and 00, ou are given the dollar value of a product in 00 and the rate at which the value of the product is epected to change during the net ears. Use this information to write a linear equation that gives the dollar value V of the product in terms of the ear t. (Let t represent 00.) 00 Value Rate 99. $0 $ decrease per ear 00. $ $.0 increase per ear Graphical Interpretation In Eercises 0 0, match the description of the situation with its graph. Also determine the slope and -intercept of each graph and interpret the slope and -intercept in the contet of the situation. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) (b) (d) 8 0. A person is paing $0 per week to a friend to repa a $00 loan. 0. An emploee is paid $8.0 per hour plus $ for each unit produced per hour. 0. A sales representative receives $0 per da for food plus $0. for each mile traveled. 0. A computer that was purchased for $70 depreciates $00 per ear. 0. Cash Flow per Share The cash flow per share for the Timberland Co. was $0.8 in 99 and $.0 in 00. Write a linear equation that gives the cash flow per share in terms of the ear. Let t represent 99. Then predict the cash flows for the ears 008 and 00. (Source: The Timberland Co.) 0. Number of Stores In 999 there were 07 J.C. Penne stores and in 00 there were 078 stores. Write a linear equation that gives the number of stores in terms of the ear. Let t 9 represent 999. Then predict the numbers of stores for the ears 008 and 00. Are our answers reasonable? Eplain. (Source: J.C. Penne Co.) Depreciation A sub shop purchases a used pizza oven for $87. After ears, the oven will have to be replaced. Write a linear equation giving the value V of the equipment during the ears it will be in use. 08. Depreciation A school district purchases a high-volume printer, copier, and scanner for $,000. After 0 ears, the equipment will have to be replaced. Its value at that time is epected to be $000. Write a linear equation giving the value V of the equipment during the 0 ears it will be in use. 09. College Enrollment The Pennslvania State Universit had enrollments of 0,7 students in 000 and,89 students in 00 at its main campus in Universit Park, Pennslvania. (Source: Penn State Fact Book) (a) Assuming the enrollment growth is linear, find a linear model that gives the enrollment in terms of the ear t, where t 0 corresponds to 000. (b) Use our model from part (a) to predict the enrollments in 008 and 00. (c) What is the slope of our model? Eplain its meaning in the contet of the situation. 0. College Enrollment The Universit of Florida had enrollments of, students in 990 and 8,7 students in 00. (Source: Universit of Florida) (a) What was the average annual change in enrollment from 990 to 00? (b) Use the average annual change in enrollment to estimate the enrollments in 99, 998, and 00. (c) Write the equation of a line that represents the given data. What is its slope? Interpret the slope in the contet of the problem. (d) Using the results of parts (a) (c), write a short paragraph discussing the concepts of slope and average rate of change.. Sales A discount outlet is offering a % discount on all items. Write a linear equation giving the sale price S for an item with a list price L.. Hourl Wage A microchip manufacturer pas its assembl line workers $.0 per hour. In addition, workers receive a piecework rate of $0.7 per unit produced. Write a linear equation for the hourl wage W in terms of the number of units produced per hour.. Cost, Revenue, and Profit A roofing contractor purchases a shingle deliver truck with a shingle elevator for $,00. The vehicle requires an average ependiture of $. per hour for fuel and maintenance, and the operator is paid $.0 per hour. (a) Write a linear equation giving the total cost C of operating this equipment for t hours. (Include the purchase cost of the equipment.)

37 (b) Assuming that customers are charged $7 per hour of machine use, write an equation for the revenue R derived from t hours of use. (c) Use the formula for profit P R C to write an equation for the profit derived from t hours of use. (d) Use the result of part (c) to find the break-even point that is, the number of hours this equipment must be used to ield a profit of 0 dollars.. Rental Demand A real estate office handles an apartment comple with 0 units. When the rent per unit is $80 per month, all 0 units are occupied. However, when the rent is $ per month, the average number of occupied units drops to 7. Assume that the relationship between the monthl rent p and the demand is linear. (a) Write the equation of the line giving the demand in terms of the rent p. (b) Use this equation to predict the number of units occupied when the rent is $. (c) Predict the number of units occupied when the rent is $9.. Geometr The length and width of a rectangular garden are meters and 0 meters, respectivel. A walkwa of width surrounds the garden. (a) Draw a diagram that gives a visual representation of the problem. (b) Write the equation for the perimeter of the walkwa in terms of. (c) Use a graphing utilit to graph the equation for the perimeter. (d) Determine the slope of the graph in part (c). For each additional one-meter increase in the width of the walkwa, determine the increase in its perimeter.. Monthl Salar A pharmaceutical salesperson receives a monthl salar of $00 plus a commission of 7% of sales. Write a linear equation for the salesperson s monthl wage W in terms of monthl sales S. 7. Business Costs A sales representative of a compan using a personal car receives $0 per da for lodging and meals plus $0.8 per mile driven. Write a linear equation giving the dail cost C to the compan in terms of, the number of miles driven. 8. Sports The median salaries (in thousands of dollars) for plaers on the Los Angeles Dodgers from 99 to 00 are shown in the scatter plot. Find the equation of the line that ou think best fits these data. (Let represent the median salar and let t represent the ear, with t corresponding to 99.) (Source: USA TODAY) Median salar (in thousands of dollars) FIGURE FOR Year ( 99) 9. Data Analsis: Cell Phone Suscribers The numbers of cellular phone suscribers (in millions) in the United States from 990 through 00, where is the ear, are shown as data points,. (Source: Cellular Telecommunications & Internet Association) (990,.) (99, 7.) (99,.0) (99,.0) (99,.) (99,.8) (99,.0) (997,.) (998, 9.) (999, 8.0) (000, 09.) (00, 8.) (00, 0.8) Model It (a) Sketch a scatter plot of the data. Let 0 correspond to 990. (b) Use a straightedge to sketch the line that ou think best fits the data. (c) Find the equation of the line from part (b). Eplain the procedure ou used. (d) Write a short paragraph eplaining the meanings of the slope and -intercept of the line in terms of the data. (e) Compare the values obtained using our model with the actual values. (f) Use our model to estimate the number of cellular phone suscribers in 008. t

38 0. Data Analsis: Average Scores An instructor gives regular 0-point quizzes and 00-point eams in an algebra course. Average scores for si students, given as data points, where is the average quiz score and is the average test score, are 8, 87, 0,, 9, 9,, 79,, 7, and, 8. [Note: There are man correct answers for parts (b) (d).] (a) Sketch a scatter plot of the data. (b) Use a straightedge to sketch the line that ou think best fits the data. (c) Find an equation for the line ou sketched in part (b). (d) Use the equation in part (c) to estimate the average test score for a person with an average quiz score of 7. (e) The instructor adds points to the average test score of each student in the class. Describe the changes in the positions of the plotted points and the change in the equation of the line. Snthesis True or False? In Eercises and, determine whether the statement is true or false. Justif our answer.. A line with a slope of 7 is steeper than a line with a slope of 7.. The line through 8, and, and the line through 0, and 7, 7 are parallel.. Eplain how ou could show that the points A,, B, 9, and C, are the vertices of a right triangle.. Eplain wh the slope of a vertical line is said to be undefined.. With the information shown in the graphs, is it possible to determine the slope of each line? Is it possible that the lines could have the same slope? Eplain. (a) (b) 8. Think About It Is it possible for two lines with positive slopes to be perpendicular? Eplain. Skills Review In Eercises 9, match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) (b) (d) In Eercises 8, find all the solutions of the equation. Check our solution(s) in the original equation The slopes of two lines are and. Which is steeper? Eplain. 7. The value V of a molding machine t ears after it is purchased is V 000t 8,00, 0 t. Eplain what the V-intercept and slope measure. 9. Make a Decision To work an etended application analzing the numbers of bachelor s degrees earned b women in the United States from 98 to 00, click on the Make a Decision button. (Data Source: U.S. Census Bureau)

39 Functions What ou should learn Determine whether relations between two variables are functions. Use function notation and evaluate functions. Find the domains of functions. Use functions to model and solve real-life problems. Evaluate difference quotients. Wh ou should learn it Functions can be used to model and solve real-life problems. For instance, in Eercise 00, ou will use a function to model the force of water against the face of a dam. Introduction to Functions Man everda phenomena involve two quantities that are related to each other b some rule of correspondence. The mathematical term for such a rule of correspondence is a relation. In mathematics, relations are often represented b mathematical equations and formulas. For instance, the simple interest I earned on $000 for ear is related to the annual interest rate r b the formula I 000r. The formula I 000r represents a special kind of relation that matches each item from one set with eactl one item from a different set. Such a relation is called a function. Definition of Function A function f from a set A to a set B is a relation that assigns to each element in the set A eactl one element in the set B. The set A is the domain (or set of inputs) of the function f, and the set B contains the range (or set of outputs). To help understand this definition, look at the function that relates the time of da to the temperature in Figure 7. Time of da (P.M.) Temperature (in degrees C) Set A is the domain. Set B contains the range. Inputs:,,,,, Outputs: 9, 0,,, FIGURE 7 This function can be represented b the following ordered pairs, in which the first coordinate ( -value) is the input and the second coordinate ( -value) is the output., 9,,,,,,,,,, 0 Video Characteristics of a Function from Set A to Set B.. Each element in A must be matched with an element in B.. Some elements in B ma not be matched with an element in A.. Two or more elements in A ma be matched with the same element in B.. An element in A (the domain) cannot be matched with two different elements in B.

40 Functions are commonl represented in four was. Four Was to Represent a Function. Verball b a sentence that describes how the input variable is related to the output variable. Numericall b a table or a list of ordered pairs that matches input values with output values. Graphicall b points on a graph in a coordinate plane in which the input values are represented b the horizontal ais and the output values are represented b the vertical ais. Algebraicall b an equation in two variables To determine whether or not a relation is a function, ou must decide whether each input value is matched with eactl one output value. If an input value is matched with two or more output values, the relation is not a function. Video Eample Testing for Functions. Determine whether the relation represents as a function of. a. The input value is the number of representatives from a state, and the output value is the number of senators. b. c. Input, Output, 0 8 FIGURE 8 a. This verbal description does describe as a function of. Regardless of the value of, the value of is alwas. Such functions are called constant functions. b. This table does not describe as a function of. The input value is matched with two different -values. c. The graph in Figure 8 does describe as a function of. Each input value is matched with eactl one output value. Now tr Eercise. Representing functions b sets of ordered pairs is common in discrete mathematics. In algebra, however, it is more common to represent functions b equations or formulas involving two variables. For instance, the equation is a function of. represents the variable as a function of the variable. In this equation, is

41 Historical Note Leonhard Euler (707 78), a Swiss mathematician, is considered to have been the most prolific and productive mathematician in histor. One of his greatest influences on mathematics was his use of smbols, or notation. The function notation f was introduced b Euler. the independent variable and is the dependent variable. The domain of the function is the set of all values taken on b the independent variable, and the range of the function is the set of all values taken on b the dependent variable. Eample Testing for Functions Represented Algebraicall Which of the equations represent(s) as a function of? a. b. To determine whether is a function of, tr to solve for in terms of. a. Solving for ields. Write original equation. Solve for. To each value of there corresponds eactl one value of. So, is a function of. b. Solving for ields ±. Write original equation. Add to each side. Solve for. The ± indicates that to a given value of there correspond two values of. So, is not a function of. Now tr Eercise. Function Notation When an equation is used to represent a function, it is convenient to name the function so that it can be referenced easil. For eample, ou know that the equation describes as a function of. Suppose ou give this function the name f. Then ou can use the following function notation. Input Output Equation f f The smbol f is read as the value of f at or simpl f of. The smbol f corresponds to the -value for a given. So, ou can write f. Keep in mind that f is the name of the function, whereas f is the value of the function at. For instance, the function given b f has function values denoted b f, f 0, f, and so on. To find these values, substitute the specified input values into the given equation. For, f. For 0, f For, f.

42 . Video Although f is often used as a convenient function name and is often used as the independent variable, ou can use other letters. For instance, f 7, f t t t 7, and g s s s 7 all define the same function. In fact, the role of the independent variable is that of a placeholder. Consequentl, the function could be described b f 7. Eample Evaluating a Function In Eample, note that g is not equal to g g. In general, g u v g u g v. Video. Let g. Find each function value. a. g b. g t c. g a. Replacing with in g ields the following. g 8 b. Replacing with t ields the following. g t t t t t c. Replacing with ields the following. g 8 8 Now tr Eercise 9. A function defined b two or more equations over a specified domain is called a piecewise-defined function. Eample A Piecewise-Defined Function Evaluate the function when, 0, and. f,, Because is less than 0, use f to obtain f. For 0, use f to obtain f 0 0. For, use f to obtain f 0. < 0 0 Now tr Eercise.

43 Technolog Use a graphing utilit to graph the functions given b and. What is the domain of each function? Do the domains of these two functions overlap? If so, for what values do the domains overlap? The Domain of a Function The domain of a function can be described eplicitl or it can be implied b the epression used to define the function. The implied domain is the set of all real numbers for which the epression is defined. For instance, the function given b f Domain ecludes -values that result in division b zero. has an implied domain that consists of all real other than ±. These two values are ecluded from the domain because division b zero is undefined. Another common tpe of implied domain is that used to avoid even roots of negative numbers. For eample, the function given b f Domain ecludes -values that result in even roots of negative numbers. is defined onl for 0. So, its implied domain is the interval 0,. In general, the domain of a function ecludes values that would cause division b zero or that would result in the even root of a negative number. Video Eample Finding the Domain of a Function.. Simulation Find the domain of each function. a. f :, 0,,, 0,,,,, b. c. Volume of a sphere: V r d. h a. The domain of f consists of all first coordinates in the set of ordered pairs. Domain,, 0,, b. Ecluding -values that ield zero in the denominator, the domain of g is the set of all real numbers ecept. c. Because this function represents the volume of a sphere, the values of the radius r must be positive. So, the domain is the set of all real numbers r such that r > 0. d. This function is defined onl for -values for which 0. Using the methods described in the Other Tpes of Inequalities section, ou can conclude that. So, the domain is the interval,. Now tr Eercise 9. In Eample (c), note that the domain of a function ma be implied b the phsical contet. For instance, from the equation V r g ou would have no reason to restrict r to positive values, but the phsical contet implies that a sphere cannot have a negative or zero radius.

44 h r = r Applications Eample The Dimensions of a Container FIGURE 9 h You work in the marketing department of a soft-drink compan and are eperimenting with a new can for iced tea that is slightl narrower and taller than a standard can. For our eperimental can, the ratio of the height to the radius is, as shown in Figure 9. a. Write the volume of the can as a function of the radius r. b. Write the volume of the can as a function of the height h. V r r h r r r a. Write V as a function of r. h h b. V h Write V as a function of h. h Now tr Eercise 87. Eample 7 The Path of a Baseball A baseball is hit at a point feet above ground at a velocit of 00 feet per second and an angle of º. The path of the baseball is given b the function f 0.00 where and are measured in feet, as shown in Figure 0. Will the baseball clear a 0-foot fence located 00 feet from home plate? Height (in feet) f() FIGURE 0 Baseball Path f() = Distance (in feet) When 00, the height of the baseball is f feet. So, the baseball will clear the fence. Now tr Eercise 9. In the equation in Eample 7, the height of the baseball is a function of the distance from home plate.

45 Number of vehicles (in thousands) FIGURE Number of Alternative-Fueled Vehicles in the U.S. V Year ( 99) t Eample 8 Alternative-Fueled Vehicles The number V (in thousands) of alternative-fueled vehicles in the United States increased in a linear pattern from 99 to 999, as shown in Figure. Then, in 000, the number of vehicles took a jump and, until 00, increased in a different linear pattern. These two patterns can be approimated b the function V t 8.08t. 8.0t 0., t 9 0 t where t represents the ear, with t corresponding to 99. Use this function to approimate the number of alternative-fueled vehicles for each ear from 99 to 00. (Source: Science Applications International Corporation; Energ Information Administration) From 99 to 999, use V t 8.08t From 000 to 00, use V t 8.0t Now tr Eercise 9. Difference Quotients One of the basic definitions in calculus emplos the ratio f h f, h h 0. This ratio is called a difference quotient, as illustrated in Eample 9.. Video Eample 9 For f 7, find f h f h Evaluating a Difference Quotient h h 7 7 h h h h 7 7 h h h h h Now tr Eercise 79. f h f. h h h h h, h 0 The smbol in calculus. indicates an eample or eercise that highlights algebraic techniques specificall used

46 You ma find it easier to calculate the difference quotient in Eample 9 b first finding f h, and then substituting the resulting epression into the difference quotient, as follows. f h h h 7 h h h 7 f h f h h h h 7 7 h h h h h h h h h, h 0 Summar of Function Terminolog Function: A function is a relationship between two variables such that to each value of the independent variable there corresponds eactl one value of the dependent variable. Function Notation: f f is the name of the function. is the dependent variable. is the independent variable. f is the value of the function at. Domain: The domain of a function is the set of all values (inputs) of the independent variable for which the function is defined. If is in the domain of f, f is said to be defined at. If is not in the domain of f, f is said to be undefined at. Range: The range of a function is the set of all values (outputs) assumed b the dependent variable (that is, the set of all function values). Implied Domain: If f is defined b an algebraic epression and the domain is not specified, the implied domain consists of all real numbers for which the epression is defined. W RITING ABOUT MATHEMATICS Everda Functions In groups of two or three, identif common real-life functions. Consider everda activities, events, and epenses, such as long distance telephone calls and car insurance. Here are two eamples. a. The statement, Your happiness is a function of the grade ou receive in this course is not a correct mathematical use of the word function. The word happiness is ambiguous. b. The statement, Your federal income ta is a function of our adjusted gross income is a correct mathematical use of the word function. Once ou have determined our adjusted gross income, our income ta can be determined. Describe our functions in words. Avoid using ambiguous words. Can ou find an eample of a piecewise-defined function?

47 Eercises The smbol Click on Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarged cop of the graph. to view the Make a Decision eercise. VOCABULARY CHECK: Fill in the blanks. Glossar. A relation that assigns to each element from a set of inputs, or, eactl one element in a set of outputs, or, is called a.. Functions are commonl represented in four different was,,,, and.. For an equation that represents as a function of, the set of all values taken on b the variable is the domain, and the set of all values taken on b the variable is the range.. The function given b f,, < 0 0 is an eample of a function.. If the domain of the function f is not given, then the set of values of the independent variable for which the epression is defined is called the. f h f. In calculus, one of the basic definitions is that of a, given b, h 0. h In Eercises, is the relationship a function?. Domain Range. Domain. Domain Range. In Eercises 8, does the table describe a function? Eplain our reasoning.. 0 National League American League 7 8 Cubs Pirates Dodgers Orioles Yankees Twins 0 Domain (Year) Input value 0 Output value Range Range (Number of North Atlantic tropical storms and hurricanes) Input value 0 0 Output value 0 Input value Output value 9 Input value 0 9 Output value In Eercises 9 and 0, which sets of ordered pairs represent functions from A to B? Eplain. 9. A 0,,, and B,, 0,, (a) 0,,,,, 0,, (b) 0,,,,,,, 0,, (c) 0, 0,, 0,, 0,, 0 (d) 0,,, 0,, 0. A a, b, c and B 0,,, (a) a,, c,, c,, b, (b) a,, b,, c, (c), a, 0, a,, c,, b (d) c, 0, b, 0, a,

48 Circulation of Newspapers In Eercises and, use the graph, which shows the circulation (in millions) of dail newspapers in the United States. (Source: Editor & Publisher Compan) Circulation (in millions) Year Morning Evening Is the circulation of morning newspapers a function of the ear? Is the circulation of evening newspapers a function of the ear? Eplain.. Let f represent the circulation of evening newspapers in ear. Find f 998. In Eercises, determine whether the equation represents as a function of In Eercises 8, evaluate the function at each specified value of the independent variable and simplif.. f (a) f (b) f (c) f. g 7 (a) g 0 (b) g 7 (c) g s 7. V r r (a) V (b) V (c) V r 8. h t t t (a) h (b) h. (c) h 9. f (a) f (b) f 0. (c) f 0. f 8 (a) f 8 (b) f (c) f 8. q 9 (a) q 0 (b) q (c). q t t t (a) q (b) q 0 (c). f (a) f (b) f (c). f (a) f (b) f (c). f, < 0, 0 (a) f (b) f 0 (c) (a) f (b) f (c) (a) f (b) f (c) In Eercises 9, complete the table f,, >, f,, (a) f (b) f (c), f 0, < <, > f f g g h t t t h t. f s s s s f s < > q q f f f f f f

49 . f, 0, > 0 f 0 f f Eploration In Eercises 7 78, match the data with one of the following functions f c, g c, h c, and r c. f 9, <, f and determine the value of the constant c that will make the function fit the data in the table In Eercises, find all real values of such that f 0.. f. f 7. f 8. f 9. f 9 0. f 8. f. f In Eercises, find the value(s) of f g.. f, g... f, f, f, g g g for which In Eercises 7 70, find the domain of the function. 7. f 8. g 9. h t 0. s t. g 0. f t t. f. f. g. h 0 7. s f s 8. f s 9. f 70. f 9 In Eercises 7 7, assume that the domain of f is the set A {,, 0,, }. Determine the set of ordered pairs that represents the function f. 7. f 7. f In Eercises 79 8, find the difference quotient and simplif our answer f, f, f, f, g, f t t, Undef f h f, h 0 h f h f, h 0 h f h f, h 0 h g g, f h f, h 0 h f t f, t t 8. f, f f, 8. f, f f 8, Geometr Write the area A of a square as a function of its perimeter P. 88. Geometr Write the area A of a circle as a function of its circumference C. The smbol in calculus. indicates an eample or eercise that highlights algebraic techniques specificall used

50 89. Maimum Volume An open bo of maimum volume is to be made from a square piece of material centimeters on a side b cutting equal squares from the corners and turning up the sides (see figure). 9. Geometr A right triangle is formed in the first quadrant b the - and -aes and a line through the point, (see figure). Write the area A of the triangle as a function of, and determine the domain of the function. (0, b) (, ) ( a, 0) 8 = (, ) (a) The table shows the volume V (in cubic centimeters) of the bo for various heights (in centimeters). Use the table to estimate the maimum volume. Height, Volume, V (b) Plot the points, V from the table in part (a). Does the relation defined b the ordered pairs represent V as a function of? (c) If V is a function of, write the function and determine its domain. 90. Maimum Profit The cost per unit in the production of a portable CD plaer is $0. The manufacturer charges $90 per unit for orders of 00 or less. To encourage large orders, the manufacturer reduces the charge b $0. per CD plaer for each unit ordered in ecess of 00 (for eample, there would be a charge of $87 per CD plaer for an order size of 0). (a) The table shows the profit P (in dollars) for various numbers of units ordered,. Use the table to estimate the maimum profit. Units, Profit, P 0 0 Units, Profit, P 7 0 (b) Plot the points, P from the table in part (a). Does the relation defined b the ordered pairs represent P as a function of? (c) If P is a function of, write the function and determine its domain. FIGURE FOR 9 FIGURE FOR 9 9. Geometr A rectangle is bounded b the -ais and the semicircle (see figure). Write the area A of the rectangle as a function of, and determine the domain of the function. 9. Path of a Ball The height (in feet) of a baseball thrown b a child is 0 where is the horizontal distance (in feet) from where the ball was thrown. Will the ball fl over the head of another child 0 feet awa tring to catch the ball? (Assume that the child who is tring to catch the ball holds a baseball glove at a height of feet.) 9. Prescription Drugs The amounts d (in billions of dollars) spent on prescription drugs in the United States from 99 to 00 (see figure) can be approimated b the model d t.0t 7, 8.7t, where t represents the ear, with t corresponding to 99. Use this model to find the amount spent on prescription drugs in each ear from 99 to 00. (Source: U.S. Centers for Medicare & Medicaid Services) Amount spent (in billions of dollars) d t 7 8 t Year ( 99) t

51 9. Average Price The average prices p (in thousands of dollars) of a new mobile home in the United States from 990 to 00 (see figure) can be approimated b the model p t where t represents the ear, with t 0 corresponding to 990. Use this model to find the average price of a mobile home in each ear from 990 to 00. (Source: U.S. Census Bureau) Mobile home price (in thousands of dollars) 0.8t 0.7t 7.,.0t., p 0 t 7 8 t Year (0 990) 9. Postal Regulations A rectangular package to be sent b the U.S. Postal Service can have a maimum combined length and girth (perimeter of a cross section) of 08 inches (see figure). t (b) Write the revenue R as a function of the number of units sold. (c) Write the profit P as a function of the number of units sold. (Note: P R C) 98. Average Cost The inventor of a new game believes that the variable cost for producing the game is $0.9 per unit and the fied costs are $000. The inventor sells each game for $.9. Let be the number of games sold. (a) The total cost for a business is the sum of the variable cost and the fied costs. Write the total cost C as a function of the number of games sold. (b) Write the average cost per unit C C as a function of. 99. Transportation For groups of 80 or more people, a charter bus compan determines the rate per person according to the formula Rate n 80, n 80 where the rate is given in dollars and n is the number of people. (a) Write the revenue R for the bus compan as a function of n. (b) Use the function in part (a) to complete the table. What can ou conclude? n R n Phsics The force F (in tons) of water against the face of a dam is estimated b the function F 9.7 0, where is the depth of the water (in feet). (a) Complete the table. What can ou conclude from the table? F (a) Write the volume V of the package as a function of. What is the domain of the function? (b) Use a graphing utilit to graph our function. Be sure to use an appropriate window setting. (c) What dimensions will maimize the volume of the package? Eplain our answer. 97. Cost, Revenue, and Profit A compan produces a product for which the variable cost is $.0 per unit and the fied costs are $98,000. The product sells for $7.98. Let be the number of units produced and sold. (a) The total cost for a business is the sum of the variable cost and the fied costs. Write the total cost C as a function of the number of units produced. (b) Use the table to approimate the depth at which the force against the dam is,000,000 tons. (c) Find the depth at which the force against the dam is,000,000 tons algebraicall. 0. Height of a Balloon A balloon carring a transmitter ascends verticall from a point 000 feet from the receiving station. (a) Draw a diagram that gives a visual representation of the problem. Let h represent the height of the balloon and let d represent the distance between the balloon and the receiving station. (b) Write the height of the balloon as a function of d. What is the domain of the function?

52 0. Wildlife The graph shows the numbers of threatened and endangered fish species in the world from 99 through 00. Let f t represent the number of threatened and endangered fish species in the ear t. (Source: U.S. Fish and Wildlife Service) f 00 f 99 (a) Find and interpret the result in the contet of the problem. (b) Find a linear model for the data algebraicall. Let N represent the number of threatened and endangered fish species and let correspond to 99. (c) Use the model found in part (b) to complete the table. N Number of threatened and endangered fish species Model It ft () Year (d) Compare our results from part (c) with the actual data. (e) Use a graphing utilit to find a linear model for the data. Let correspond to 99. How does the model ou found in part (b) compare with the model given b the graphing utilit? t Snthesis True or False? In Eercises 0 and 0, determine whether the statement is true or false. Justif our answer. 0. The domain of the function given b f is,, and the range of f is 0,. 0. The set of ordered pairs 8,,, 0,, 0,,, 0,,, represents a function. 0. Writing In our own words, eplain the meanings of domain and range. 0. Think About It Consider f and g. Wh are the domains of f and g different? In Eercises 07 and 08, determine whether the statements use the word function in was that are mathematicall correct. Eplain our reasoning. 07. (a) The sales ta on a purchased item is a function of the selling price. (b) Your score on the net algebra eam is a function of the number of hours ou stud the night before the eam. 08. (a) The amount in our savings account is a function of our salar. (b) The speed at which a free-falling baseball strikes the ground is a function of the height from which it was dropped. Skills Review In Eercises 09, solve the equation t t t t 9 In Eercises, find the equation of the line passing through the pair of points..,,,.,.,,,.,,, 0, 0, 9

53 Analzing Graphs of Functions What ou should learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals on which functions are increasing or decreasing and determine relative maimum and relative minimum values of functions. Determine the average rate of change of a function. Identif even and odd functions. Wh ou should learn it Graphs of functions can help ou visualize relationships between variables in real life. For instance, in Eercise 8, ou will use the graph of a function to represent visuall the temperature for a cit over a hour period. The Graph of a Function In the previous section, ou studied functions from an algebraic point of view. In this section, ou will stud functions from a graphical perspective. The graph of a function f is the collection of ordered pairs, f such that is in the domain of f. As ou stud this section, remember that the directed distance from the -ais f the directed distance from the -ais as shown in Figure. f() = f() Video FIGURE. Eample Finding the Domain and Range of a Function (, ) Range FIGURE = f( ) (0, ) (, ) (, ) Domain Use the graph of the function f, shown in Figure, to find (a) the domain of f, (b) the function values f and f, and (c) the range of f. a. The closed dot at, indicates that is in the domain of f, whereas the open dot at, indicates that is not in the domain. So, the domain of f is all in the interval,. b. Because, is a point on the graph of f, it follows that f. Similarl, because, is a point on the graph of f, it follows that f. c. Because the graph does not etend below f or above f 0, the range of f is the interval,. Now tr Eercise. The use of dots (open or closed) at the etreme left and right points of a graph indicates that the graph does not etend beond these points. If no such dots are shown, assume that the graph etends beond these points.

54 B the definition of a function, at most one -value corresponds to a given -value. This means that the graph of a function cannot have two or more different points with the same -coordinate, and no two points on the graph of a function can be verticall above or below each other. It follows, then, that a vertical line can intersect the graph of a function at most once. This observation provides a convenient visual test called the Vertical Line Test for functions. Vertical Line Test for Functions A set of points in a coordinate plane is the graph of as a function of if and onl if no vertical line intersects the graph at more than one point. Video Eample Vertical Line Test for Functions. Use the Vertical Line Test to decide whether the graphs in Figure represent as a function of. (a) FIGURE (b) (c) a. This is not a graph of as a function of, because ou can find a vertical line that intersects the graph twice. That is, for a particular input, there is more than one output. b. This is a graph of as a function of, because ever vertical line intersects the graph at most once. That is, for a particular input, there is at most one output. c. This is a graph of as a function of. (Note that if a vertical line does not intersect the graph, it simpl means that the function is undefined for that particular value of. ) That is, for a particular input, there is at most one output. Now tr Eercise 9.

55 .. Simulation f ( ) = + 0 (, 0) (, 0) Zeros of f: FIGURE 8, 8 g ( ) = 0 ( 0, 0) 0, 0 Zeros of g: FIGURE ± 0 (, 0) ( ) ht () = t t + t Zeros of a Function If the graph of a function of has an -intercept at a, 0, then a is a zero of the function. Zeros of a Function The zeros of a function f of are the -values for which f 0. Video Eample Find the zeros of each function. Finding the Zeros of a Function a. f 0 b. g 0 c. h t To find the zeros of a function, set the function equal to zero and solve for the independent variable. a. 0 0 Set f equal to Factor. Set st factor equal to 0. Set nd factor equal to 0. The zeros of f are and In Figure, note that the graph of has, 0. f and, 0 as its -intercepts. b ± 0 Set g equal to 0. Square each side. Add to each side. Etract square roots. The zeros of g are 0 and 0. In Figure, note that the graph of g has 0, 0 and 0, 0 as its -intercepts. c. t t 0 Set h t equal to 0. t 0 t Multipl each side b t. Add to each side. t t 8 t Divide each side b. Zero of h: FIGURE 7 t The zero of h is t. In Figure 7, note that the graph of h has as its t-intercept. Now tr Eercise., 0

56 Decreasing FIGURE 8 Constant Increasing Increasing and Decreasing Functions The more ou know about the graph of a function, the more ou know about the function itself. Consider the graph shown in Figure 8. As ou move from left to right, this graph falls from to 0, is constant from 0 to, and rises from to. Increasing, Decreasing, and Constant Functions A function f is increasing on an interval if, for an and in the interval, < implies f < f. A function f is decreasing on an interval if, for an and in the interval, < implies f > f. A function f is constant on an interval if, for an and in the interval, f f.. Video Eample Increasing and Decreasing Functions Use the graphs in Figure 9 to describe the increasing or decreasing behavior of each function. a. This function is increasing over the entire real line. b. This function is increasing on the interval,, decreasing on the interval,, and increasing on the interval,. c. This function is increasing on the interval, 0, constant on the interval 0,, and decreasing on the interval,. f() = f() = (, ) (0, ) (, ) t (, ) f(t) = t +, t < 0, 0 t t +, t > (a) FIGURE 9 (b) (c) Now tr Eercise. To help ou decide whether a function is increasing, decreasing, or constant on an interval, ou can evaluate the function for several values of. However, calculus is needed to determine, for certain, all intervals on which a function is increasing, decreasing, or constant.

57 A relative minimum or relative maimum is also referred to as a local minimum or local maimum. Relative maima The points at which a function changes its increasing, decreasing, or constant behavior are helpful in determining the relative minimum or relative maimum values of the function. Definitions of Relative Minimum and Relative Maimum A function value f a is called a relative minimum of f if there eists an interval, that contains a such that < < implies f a f. A function value f a is called a relative maimum of f if there eists an interval, that contains a such that < < implies f a f.. FIGURE 0 Relative minima FIGURE Video f ( ) = Figure 0 shows several different eamples of relative minima and relative maima. In the Quadratic Functions and Models section, ou will stud a technique for finding the eact point at which a second-degree polnomial function has a relative minimum or relative maimum. For the time being, however, ou can use a graphing utilit to find reasonable approimations of these points. Eample Approimating a Relative Minimum Use a graphing utilit to approimate the relative minimum of the function given b f. The graph of f is shown in Figure. B using the zoom and trace features or the minimum feature of a graphing utilit, ou can estimate that the function has a relative minimum at the point 0.7,.. Relative minimum Later, in the Quadratic Functions and Models section, ou will be able to determine that the eact point at which the relative minimum occurs is, 0. Now tr Eercise 9. You can also use the table feature of a graphing utilit to approimate numericall the relative minimum of the function in Eample. Using a table that begins at 0. and increments the value of b 0.0, ou can approimate that the minimum of f occurs at the point 0.7,.. Technolog If ou use a graphing utilit to estimate the - and -values of a relative minimum or relative maimum, the zoom feature will often produce graphs that are nearl flat. To overcome this problem, ou can manuall change the vertical setting of the viewing window. The graph will stretch verticall if the values of Ymin and Yma are closer together.

58 FIGURE (, f( )) f(). = (, f( )) Secant line f f( ) f( ) FIGURE Video (0, 0) (, ) Eploration (, ) Use the information in Eample 7 to find the average speed of the car from t 0 to t 9 seconds. Eplain wh the result is less than the value obtained in part (b). Average Rate of Change In the Linear Equations in Two Variables section, ou learned that the slope of a line can be interpreted as a rate of change. For a nonlinear graph whose slope changes at each point, the average rate of change between an two points, f and, f is the slope of the line through the two points (see Figure ). The line through the two points is called the secant line, and the slope of this line is denoted as m sec. Average rate of change of f from to Eample f f Average Rate of Change of a Function b. The average rate of change of f from 0 to is f f Now tr Eercise. Finding Average Speed The distance s (in feet) a moving car is from a stoplight is given b the function s t 0t, where t is the time (in seconds). Find the average speed of the car (a) from t 0 to t seconds and (b) from t to t 9 seconds. a. The average speed of the car from t 0 to t seconds is s t s t t t 0 feet per second. b. The average speed of the car from t to t 9 seconds is s t s t t t f 0 f 0 f f 0 0 s s 0 0 s 9 s 9 Now tr Eercise f f m sec 0 0 change in change in Find the average rates of change of f (a) from to 0 and (b) from 0 to (see Figure ). a. The average rate of change of f from to 0 is Secant line has positive slope. Eample 7.. Secant line has negative slope. 7 feet per second.

59 Even and Odd Functions In the Graphs of Equations section, ou studied different tpes of smmetr of a graph. In the terminolog of functions, a function is said to be even if its graph is smmetric with respect to the -ais and to be odd if its graph is smmetric with respect to the origin. The smmetr tests in the Graphs of Equations section ield the following tests for even and odd functions.. Eploration Graph each of the functions with a graphing utilit. Determine whether the function is even, odd, or neither. f g h j 8 k p 9 What do ou notice about the equations of functions that are odd? What do ou notice about the equations of functions that are even? Can ou describe a wa to identif a function as odd or even b inspecting the equation? Can ou describe a wa to identif a function as neither odd nor even b inspecting the equation? Tests for Even and Odd Functions A function f is even if, for each in the domain of f, f f. A function f is odd if, for each in the domain of f, f f. Video Video Video Eample 8 Even and Odd Functions a. The function g is odd because g g, as follows. g g Substitute for. Simplif. Distributive Propert Test for odd function b. The function h is even because h h, as follows. h h Substitute for. Simplif. Test for even function The graphs and smmetr of these two functions are shown in Figure. g() = (, ) (, ) (, ) (, ) h() = + (a) Smmetric to origin: Odd Function FIGURE Now tr Eercise 7. (b) Smmetric to -ais: Even Function

60 Eercises The smbol Click on Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarged cop of the graph. to view the Make a Decision eercise. VOCABULARY CHECK: Glossar Fill in the blanks.. The graph of a function f is the collection of or, f such that is in the domain of f.. The is used to determine whether the graph of an equation is a function of in terms of.. The of a function f are the values of for which f 0.. A function f is on an interval if, for an and in the interval, < implies f > f.. A function value f a is a relative of f if there eists an interval, containing a such that < < implies f a f.. The between an two points, f and, f is the slope of the line through the two points, and this line is called the line. 7. A function f is if for the each in the domain of f, f f. 8. A function f is if its graph is smmetric with respect to the -ais. In Eercises, use the graph of the function to find the domain and range of f..... In Eercises 8, use the graph of the function to find the indicated function values.. (a) f (b) f. (a) f (b) f (c) (d) f (c) f 0 (d) f = f() f = f() = f() = f() = f() = f() 7. (a) f (b) f 8. (a) f (b) f (c) f 0 (d) f (c) f (d) f In Eercises 9, use the Vertical Line Test to determine whether is a function of = f().. = f()

61 ... f. f In Eercises, find the zeros of the function algebraicall.. f 7 0. f f 9 f f f 9 f f 9 f f 8.., f,, (0, ) (, ) f,, 0 0 < > > (, 0) (, 0) In Eercises 0, (a) use a graphing utilit to graph the function and find the zeros of the function and (b) verif our results from part (a) algebraicall f. f 7 7. f 8. f 8 f f 9 In Eercises 8, determine the intervals over which the function is increasing, decreasing, or constant.. f. f f (, ) (, ) f (0, ) (, ) (, )

62 In Eercises 9 8, (a) use a graphing utilit to graph the function and visuall determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verif whether the function is increasing, decreasing, or constant over the intervals ou identified in part (a). 9. f 0. g. g s s. h. f t t. f. f. f 7. f 8. f In Eercises 9, use a graphing utilit to graph the function and approimate (to two decimal places) an relative minimum or relative maimum values. 9. f 0. f. f. f 9.. f f In Eercises, graph the function and determine the interval(s) for which f 0.. f. f 7. f 8. f 9. f 0. f. f. In Eercises 70, find the average rate of change of the function from to. Function. f. f( 8. f. f 8 7. f 8. f 9. f 70. f f -Values 0, 0,,,,,,, 8 In Eercises 7 7, determine whether the function is even, odd, or neither. Then describe the smmetr. 7. f 7. h 7. g 7. f 7. f t t t 7. g s s In Eercises 77 80, write the height h of the rectangle as a function of = = 80. (, ) h In Eercises 8 8, write the length L of the rectangle as a function of (, ) (, ) L = = = L (8, ) 8 (, ) 8. Electronics The number of lumens (time rate of flow of light) L from a fluorescent lamp can be approimated b the model L , h = (, ) (, ) = (, ) L 0 90 where is the wattage of the lamp. (a) Use a graphing utilit to graph the function. (b) Use the graph from part (a) to estimate the wattage necessar to obtain 000 lumens. h (, ) L h = (8, ) 8 =

63 8. Data Analsis: Temperature The table shows the temperature (in degrees Fahrenheit) of a certain cit over a -hour period. Let represent the time of da, where 0 corresponds to A.M. Time, Model It Temperature, A model that represents these data is given b , 0. (a) Use a graphing utilit to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? (c) Use the graph to approimate the times when the temperature was increasing and decreasing. (d) Use the graph to approimate the maimum and minimum temperatures during this -hour period. (e) Could this model be used to predict the temperature for the cit during the net -hour period? Wh or wh not? 87. Coordinate Ais Scale Each function models the specified data for the ears 99 through 00, with t corresponding to 99. Estimate a reasonable scale for the vertical ais (e.g., hundreds, thousands, millions, etc.) of the graph and justif our answer. (There are man correct answers.) (a) f t represents the average salar of college professors. (b) f t represents the U.S. population. (c) f t represents the percent of the civilian work force that is unemploed. 88. Geometr Corners of equal size are cut from a square with sides of length 8 meters (see figure). (a) Write the area A of the resulting figure as a function of. Determine the domain of the function. (b) Use a graphing utilit to graph the area function over its domain. Use the graph to find the range of the function. (c) Identif the figure that would result if were chosen to be the maimum value in the domain of the function. What would be the length of each side of the figure? 89. Digital Music Sales The estimated revenues r (in billions of dollars) from sales of digital music from 00 to 007 can be approimated b the model r.9t 0.7t 0.t 0, where t represents the ear, with t corresponding to 00. (Source: Fortune) (a) Use a graphing utilit to graph the model. (b) Find the average rate of change of the model from 00 to 007. Interpret our answer in the contet of the problem. 90. Foreign College Students The numbers of foreign students F (in thousands) enrolled in colleges in the United States from 99 to 00 can be approimated b the model. F 0.00t 0.t., t t 7 where t represents the ear, with t corresponding to 99. (Source: Institute of International Education) (a) Use a graphing utilit to graph the model. (b) Find the average rate of change of the model from 99 to 00. Interpret our answer in the contet of the problem. (c) Find the five-ear time periods when the rate of change was the greatest and the least. 8 8

64 Phsics In Eercises 9 9, (a) use the position equation s t v 0 t s 0 to write a function that represents the situation, (b) use a graphing utilit to graph the function, (c) find the average rate of change of the function from t to t, (d) interpret our answer to part (c) in the contet of the problem, (e) find the equation of the secant line through t and t, and (f) graph the secant line in the same viewing window as our position function. 9. An object is thrown upward from a height of feet at a velocit of feet per second. t 0, t 9. An object is thrown upward from a height of. feet at a velocit of 7 feet per second. t 0, t 9. An object is thrown upward from ground level at a velocit of 0 feet per second. t, t 9. An object is thrown upward from ground level at a velocit of 9 feet per second. t, t 9. An object is dropped from a height of 0 feet. t 0, t 9. An object is dropped from a height of 80 feet. t, t Snthesis True or False? In Eercises 97 and 98, determine whether the statement is true or false. Justif our answer. 97. A function with a square root cannot have a domain that is the set of real numbers. 98. It is possible for an odd function to have the interval 0, as its domain. 99. If f is an even function, determine whether g is even, odd, or neither. Eplain. (a) g f (b) g f (c) g f (d) g f 00. Think About It Does the graph in Eercise represent as a function of? Eplain. Think About It In Eercises 0 0, find the coordinates of a second point on the graph of a function f if the given point is on the graph and the function is (a) even and (b) odd , 9 0.,,, 7 0. Writing Use a graphing utilit to graph each function. Write a paragraph describing an similarities and differences ou observe among the graphs. (a) (b) (c) (d) (e) (f) 0. Conjecture Use the results of Eercise 0 to make a conjecture about the graphs of 7 and 8. Use a graphing utilit to graph the functions and compare the results with our conjecture. Skills Review In Eercises 07 0, solve the equation In Eercises, evaluate the function at each specified value of the independent variable and simplif.. f 8 (a) f 9 (b) f (c) f 7. f 0 (a) f (b) f 8 (c) f. f 9 (a) f (b) f 0 (c) f. f (a) f (b) f (c) f In Eercises and, find the difference quotient and simplif our answer. f h f. f 9,, h 0 h f h f. f,, h 0 h

65 A Librar of Functions What ou should learn Identif and graph linear and squaring functions. Identif and graph cubic, square root, and reciprocal functions. Identif and graph step and other piecewise-defined functions. Recognize graphs of parent functions. Wh ou should learn it Step functions can be used to model real-life situations. For instance, in Eercise, ou will use a step function to model the cost of sending an overnight package from Los Angeles to Miami. Linear and Squaring Functions One of the goals of this tet is to enable ou to recognize the basic shapes of the graphs of different tpes of functions. For instance, ou know that the graph of the linear function f a b is a line with slope m a and -intercept at 0, b. The graph of the linear function has the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all real numbers. The graph has an -intercept of b m, 0 and a -intercept of 0, b. The graph is increasing if m > 0, decreasing if m < 0, and constant if m 0. Eample Writing a Linear Function Write the linear function f for which f and f 0. To find the equation of the line that passes through,,,, 0, first find the slope of the line. m Net, use the point-slope form of the equation of a line. m f 0 Point-slope form Substitute for,, and m. Simplif. Function notation The graph of this function is shown in Figure. and f() = + FIGURE Now tr Eercise.

66 There are two special tpes of linear functions, the constant function and the identit function. A constant function has the form f c and has the domain of all real numbers with a range consisting of a single real number c. The graph of a constant function is a horizontal line, as shown in Figure. The identit function has the form f. Its domain and range are the set of all real numbers. The identit function has a slope of m and a -intercept 0, 0. The graph of the identit function is a line for which each -coordinate equals the corresponding -coordinate. The graph is alwas increasing, as shown in Figure 7. f() = f() = c. Video FIGURE FIGURE 7 The graph of the squaring function f is a U-shaped curve with the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all nonnegative real numbers. The function is even. The graph has an intercept at 0, 0. The graph is decreasing on the interval, 0 and increasing on the interval 0,. The graph is smmetric with respect to the -ais. The graph has a relative minimum at 0, 0. The graph of the squaring function is shown in Figure 8. f() = (0, 0) FIGURE 8

67 .. Video Video Cubic, Square Root, and Reciprocal Functions The basic characteristics of the graphs of the cubic, square root, and reciprocal functions are summarized below.. The graph of the cubic function f has the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all real numbers. The function is odd. The graph has an intercept at 0, 0. The graph is increasing on the interval,. The graph is smmetric with respect to the origin. The graph of the cubic function is shown in Figure 9.. The graph of the square root function f has the following characteristics. The domain of the function is the set of all nonnegative real numbers. The range of the function is the set of all nonnegative real numbers. The graph has an intercept at 0, 0. The graph is increasing on the interval 0,. The graph of the square root function is shown in Figure 70.. The graph of the reciprocal function f has the following characteristics. The domain of the function is, 0 0,. The range of the function is, 0 0,. The function is odd. The graph does not have an intercepts. The graph is decreasing on the intervals, 0 and 0,. The graph is smmetric with respect to the origin. The graph of the reciprocal function is shown in Figure 7. f() = f() = f() = (0, 0) (0, 0) Cubic function FIGURE 9 Square root function FIGURE 70 Reciprocal function FIGURE 7

68 FIGURE 7 f () =[[ ]] Technolog When graphing a step function, ou should set our graphing utilit to dot mode. Step and Piecewise-Defined Functions Functions whose graphs resemble sets of stairsteps are known as step functions. The most famous of the step functions is the greatest integer function, which is denoted b and defined as f the greatest integer less than or equal to. Some values of the greatest integer function are as follows. greatest integer greatest integer 0 greatest integer 0 0. greatest integer. The graph of the greatest integer function f has the following characteristics, as shown in Figure 7. The domain of the function is the set of all real numbers. The range of the function is the set of all integers. The graph has a -intercept at 0, 0 and -intercepts in the interval 0,. The graph is constant between each pair of consecutive integers. The graph jumps verticall one unit at each integer value. Eample Evaluating a Step Function FIGURE 7 f ()=[[ ]] + Evaluate the function when,, and f For, the greatest integer is, so f 0. For, the greatest integer is, so f. For, the greatest integer is, so f. You can verif our answers b eamining the graph of f shown in Figure 7. Now tr Eercise 9. Recall from the Functions section that a piecewise-defined function is defined b two or more equations over a specified domain. To graph a piecewisedefined function, graph each equation separatel over the specified domain, as shown in Eample..

69 . = + FIGURE 7 Video = + Eample Sketch the graph of f,, Graphing a Piecewise-Defined Function This piecewise-defined function is composed of two linear functions. At and to the left of the graph is the line, and to the right of the graph is the line, as shown in Figure 7. Notice that the point, is a solid dot and the point, is an open dot. This is because f. Parent Functions >. Now tr Eercise. The eight graphs shown in Figure 7 represent the most commonl used functions in algebra. Familiarit with the basic characteristics of these simple graphs will help ou analze the shapes of more complicated graphs in particular, graphs obtained from these graphs b the rigid and nonrigid transformations studied in the net section. f() = f() = f() = c f() = (a) Constant Function (b) Identit Function (c) Absolute Value Function (d) Square Root Function f() = f() = f() = f () =[[ ]] (e) Quadratic Function FIGURE 7 (f) Cubic Function (g) Reciprocal Function (h) Greatest Integer Function

70 Eercises The smbol indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. Click on to view the complete solution of the eercise. Click on to print an enlarged cop of the graph. Click on to view the Make a Decision eercise. VOCABULARY CHECK: Match each function with its name. Glossar. f. f. f. f. f. f c f 9. f a b f (a) squaring function (b) square root function (c) cubic function (d) linear function (e) constant function (f) absolute value function (e) greatest integer function (h) reciprocal function (i) identit function In Eercises 8, (a) write the linear function f such that it has the indicated function values and (b) sketch the graph of the function.. f, f 0. f 8, f. f, f 7. f 9, f.. f, f f 0, f 7. f, f 8. f, f In Eercises 9 8, use a graphing utilit to graph the function. Be sure to choose an appropriate viewing window. 9. f 0. f. f. f. f. f 8. h. g 7. f 8. f 8 9. f 0. g. f. f. g. h. f. f 7. h 8. k In Eercises 9, evaluate the function for the indicated values. 9. f (a) f. (b) f.9 (c) f. (d) 0. g (a) g (b) g 0. (c) g 9. (d) f 7 g. h (a) h (b) h (c) h. (d) h.. f 7 (a) f 0 (b) f. (c) f (d) f. h (a) h. (b) h. (c) h 7 (d) h. k (a) k (b) k. (c) k 0. (d) k. g (a) g.7 (b) g (c) g 0.8 (d) g.. g 7 (a) (b) g 9 (c) g (d) In Eercises 7, sketch the graph of the function. 7. g 8. g 9. g 0. g. g. g In Eercises 0, graph the function..... g 8 f,, g,, > f,, < 0 0 f,, < f,, > > g

71 h,, h,,,, k,, In Eercises and, (a) use a graphing utilit to graph the function, (b) state the domain and range of the function, and (c) describe the pattern of the graph. s.. In Eercises 0, (a) identif the parent function and the transformed parent function shown in the graph, (b) write an equation for the function shown in the graph, and (c) use a graphing utilit to verif our answers in parts (a) and (b) < 0 0 < < 0 0 < > g. Communications The cost of a telephone call between Denver and Boise is $0.0 for the first minute and $0. for each additional minute or portion of a minute. A model for the total cost C (in dollars) of the phone call is C t, t > 0 where t is the length of the phone call in minutes. (a) Sketch the graph of the model. (b) Determine the cost of a call lasting minutes and 0 seconds.. Communications The cost of using a telephone calling card is $.0 for the first minute and $0.8 for each additional minute or portion of a minute. (a) A customer needs a model for the cost C of using a calling card for a call lasting t minutes. Which of the following is the appropriate model? Eplain. C t t C t t (b) Graph the appropriate model. Determine the cost of a call lasting 8 minutes and seconds.. Deliver Charges The cost of sending an overnight package from Los Angeles to Miami is $0.7 for a package weighing up to but not including pound and $.9 for each additional pound or portion of a pound. A model for the total cost C (in dollars) of sending the package is C 0.7.9, > 0 where is the weight in pounds. (a) Sketch a graph of the model. (b) Determine the cost of sending a package that weighs 0. pounds.. Deliver Charges The cost of sending an overnight package from New York to Atlanta is $9.80 for a package weighing up to but not including pound and $.0 for each additional pound or portion of a pound. (a) Use the greatest integer function to create a model for the cost C of overnight deliver of a package weighing pounds, > 0. (b) Sketch the graph of the function.. Wages A mechanic is paid $.00 per hour for regular time and time-and-a-half for overtime. The weekl wage function is given b W h h, 8 h 0 80, 0 < h 0 h > 0 where h is the number of hours worked in a week. (a) Evaluate W 0, W 0, W, and W 0. (b) The compan increased the regular work week to hours. What is the new weekl wage function?

72 . Snowstorm During a nine-hour snowstorm, it snows at a rate of inch per hour for the first hours, at a rate of inches per hour for the net hours, and at a rate of 0. inch per hour for the final hour. Write and graph a piecewise-defined function that gives the depth of the snow during the snowstorm. How man inches of snow accumulated from the storm? Model It 7. Revenue The table shows the monthl revenue (in thousands of dollars) of a landscaping business for each month of the ear 00, with representing Januar. Volume (in gallons) V (0, 00) 00 (0, 7) (0, 7) 7 (, 0) 0 (, 0) (0, 0) (0, ) (0, ) (0, 0) Time (in minutes) FIGURE FOR 8 t Month, Revenue, Snthesis A mathematical model that represents these data is f (a) What is the domain of each part of the piecewisedefined function? How can ou tell? Eplain our reasoning. (b) Sketch a graph of the model. (c) Find f and f, and interpret our results in the contet of the problem. (d) How do the values obtained from the model in part (b) compare with the actual data values? 8. Fluid Flow The intake pipe of a 00-gallon tank has a flow rate of 0 gallons per minute, and two drainpipes have flow rates of gallons per minute each. The figure shows the volume V of fluid in the tank as a function of time t. Determine the combination of the input pipe and drain pipes in which the fluid is flowing in specific subintervals of the hour of time shown on the graph. (There are man correct answers.) True or False? In Eercises 9 and 70, determine whether the statement is true or false. Justif our answer. 9. A piecewise-defined function will alwas have at least one -intercept or at least one -intercept. 70. f,,, can be rewritten as f, <. Eploration In Eercises 7 and 7, write equations for the piecewise-defined function shown in the graph (0, ) (, ) 8 Skills Review < < < (8, 0) In Eercises 7 and 7, solve the inequalit and sketch the solution on the real number line > 9 L In Eercises 7 and 7, determine whether the lines and L passing through the pairs of points are parallel, perpendicular, or neither. 7. L :,,, 0 7. L :, 7,, L :,,, 9 L :,,, (, ) (, ) (7, 0) (, ) (0, 0)

73 Transformations of Functions What ou should learn Use vertical and horizontal shifts to sketch graphs of functions. Use reflections to sketch graphs of functions. Use nonrigid transformations to sketch graphs of functions. Wh ou should learn it Knowing the graphs of common functions and knowing how to shift, reflect, and stretch graphs of functions can help ou sketch a wide variet of simple functions b hand. This skill is useful in sketching graphs of functions that model real-life data, such as in Eercise 8, where ou are asked to sketch the graph of a function that models the amounts of mortgage debt outstanding from 990 through 00. Shifting Graphs Man functions have graphs that are simple transformations of the parent graphs summarized in the previous section. For eample, ou can obtain the graph of h b shifting the graph of f upward two units, as shown in Figure 7. In function notation, h and f are related as follows. h f Similarl, ou can obtain the graph of g Upward shift of two units b shifting the graph of f to the right two units, as shown in Figure 77. In this case, the functions g and f have the following relationship. g f h() = + Right shift of two units f() = g() = ( ) f() = FIGURE 7 FIGURE 77 The following list summarizes this discussion about horizontal and vertical shifts.. Simulation In items and, be sure ou see that h f c corresponds to a right shift and h f c corresponds to a left shift for c > 0. Vertical and Horizontal Shifts Let c be a positive real number. Vertical and horizontal shifts in the graph of f are represented as follows.. Vertical shift c units upward: h f c. Vertical shift c units downward: h f c. Horizontal shift c units to the right: h f c. Horizontal shift c units to the left: h f c

74 Some graphs can be obtained from combinations of vertical and horizontal shifts, as demonstrated in Eample (b). Vertical and horizontal shifts generate a famil of functions, each with the same shape but at different locations in the plane. Video. Video. Eample Shifts in the Graphs of a Function Use the graph of f to sketch the graph of each function. a. g b. h a. Relative to the graph of f, the graph of g is a downward shift of one unit, as shown in Figure 78. b. Relative to the graph of f, the graph of h involves a left shift of two units and an upward shift of one unit, as shown in Figure 79. f () = h() = ( + ) + f() = g () = FIGURE 78 Now tr Eercise. FIGURE79 In Figure 79, notice that the same result is obtained if the vertical shift precedes the horizontal shift or if the horizontal shift precedes the vertical shift. Graphing utilities are ideal tools for eploring translations of functions. Graph f, g, and h in same viewing window. Before looking at the graphs, tr to predict how the graphs of g and h relate to the graph of f. a. b. Eploration f, g, h f, g, h c. f, g, h

75 f() = h() = Reflecting Graphs The second common tpe of transformation is a reflection. For instance, if ou consider the -ais to be a mirror, the graph of h is the mirror image (or reflection) of the graph of f, as shown in Figure 80. FIGURE 80 Reflections in the Coordinate Aes Reflections in the coordinate aes of the graph of f are represented as follows.. Reflection in the -ais: h f f () =. Reflection in the -ais: h f Eample Finding Equations from Graphs FIGURE 8 The graph of the function given b f is shown in Figure 8. Each of the graphs in Figure 8 is a transformation of the graph of f. Find an equation for each of these functions. Video. = g( ) = h( ) Eploration Reverse the order of transformations in Eample (a). Do ou obtain the same graph? Do the same for Eample (b). Do ou obtain the same graph? Eplain. (a) (b) FIGURE 8 a. The graph of g is a reflection in the -ais followed b an upward shift of two units of the graph of f. So, the equation for g is g. b. The graph of h is a horizontal shift of three units to the right followed b a reflection in the -ais of the graph of f. So, the equation for h is h. Now tr Eercise 9.

76 Eample Reflections and Shifts Video Compare the graph of each function with the graph of f. a.. g b. h c. k Algebraic Graphical a. The graph of g is a reflection of the graph of f in the -ais because g f. b. The graph of h is a reflection of the graph of f in the -ais because h f. c. The graph of k is a left shift of two units followed b a reflection in the -ais because k f. a. Graph f and g on the same set of coordinate aes. From the graph in Figure 8, ou can see that the graph of g is a reflection of the graph of f in the -ais. b. Graph f and h on the same set of coordinate aes. From the graph in Figure 8, ou can see that the graph of h is a reflection of the graph of f in the -ais. c. Graph f and k on the same set of coordinate aes. From the graph in Figure 8, ou can see that the graph of k is a left shift of two units of the graph of f, followed b a reflection in the -ais. f() = h() = g() = f() = FIGURE 8 FIGURE 8 f () = k ( ) = + Now tr Eercise 9. FIGURE 8 When sketching the graphs of functions involving square roots, remember that the domain must be restricted to eclude negative numbers inside the radical. For instance, here are the domains of the functions in Eample. Domain of g : 0 Domain of h : 0 Domain of k :

77 . FIGURE 8 g() = f() = h() = f() = FIGURE 88 f() = FIGURE 87 Video g() = 8 h() = 8 Nonrigid Transformations Horizontal shifts, vertical shifts, and reflections are rigid transformations because the basic shape of the graph is unchanged. These transformations change onl the position of the graph in the coordinate plane. Nonrigid transformations are those that cause a distortion a change in the shape of the original graph. For instance, a nonrigid transformation of the graph of f is represented b g cf, where the transformation is a vertical stretch if c > and a vertical shrink if 0 < c <. Another nonrigid transformation of the graph of f is represented b h f c, where the transformation is a horizontal shrink if c > and a horizontal stretch if 0 < c <. Eample Nonrigid Transformations Compare the graph of each function with the graph of h a. b. a. Relative to the graph of f, the graph of h is a vertical stretch (each -value is multiplied b ) of the graph of f. (See Figure 8.) b. Similarl, the graph of g g f f is a vertical shrink each -value is multiplied b of the graph of f. (See Figure 87.) Now tr Eercise. Eample Nonrigid Transformations Compare the graph of each function with the graph of f. a. g f b. h f a. Relative to the graph of f, the graph of g f 8 f. is a horizontal shrink c > of the graph of f. (See Figure 88.) b. Similarl, the graph of f() = FIGURE 89 h f 8 is a horizontal stretch 0 < c < of the graph of f. (See Figure 89.) Now tr Eercise 7.

78 Eercises The smbol Click on Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarged cop of the graph. to view the Make a Decision eercise. VOCABULARY CHECK: In Eercises, fill in the blanks. Glossar. Horizontal shifts, vertical shifts, and reflections are called transformations.. A reflection in the -ais of f is represented b h, while a reflection in the -ais of f is represented b h.. Transformations that cause a distortion in the shape of the graph of f are called transformations.. A nonrigid transformation of f represented b h f c is a if c > and a if 0 < c <.. A nonrigid transformation of f represented b g cf is a if c > and a if 0 < c <.. Match the rigid transformation of f with the correct representation of the graph of h, where c > 0. (a) h f c (i) A horizontal shift of f, c units to the right (b) h f c (ii) A vertical shift of f, c units downward (c) h f c (iii) A horizontal shift of f, c units to the left (d) h f c (iv) A vertical shift of f, c units upward. For each function, sketch (on the same set of coordinate aes) a graph of each function for c,, and. (a) f c (b) f c (c) f c. For each function, sketch (on the same set of coordinate aes) a graph of each function for c,,, and. (a) f c (b) f c (c) f c. For each function, sketch (on the same set of coordinate aes) a graph of each function for c, 0, and. (a) f c (b) f c (c) f c. For each function, sketch (on the same set of coordinate aes) a graph of each function for c,,, and. (a) (b) f c, c, f c, c, < 0 0 < 0 0 In Eercises 8, use the graph of f to sketch each graph.. (a) f. (a) f (b) f (b) f (c) f (c) f (d) f (d) f (e) f (e) f (f) f (f) f (g) f (g) f (, 0) (, ) (, ) f (0, ) (, ) (, ) f (, ) 8 (0, ) FIGURE FOR FIGURE FOR 7. (a) f 8. (a) f (b) f (b) f (c) f (c) f (d) f (d) f (e) f (e) f (f) f (f) (g) f 0 f (g) f 8

79 (, ) f (0, ) (, 0) (, ) (0, ) (, 0) (, 0) 0 f (, ) (, ) 0. Use the graph of to write an equation for each function whose graph is shown. f (a) (b) FIGURE FOR 7 FIGURE FOR 8 9. Use the graph of f to write an equation for each function whose graph is shown. (a) (b) (c) 0. Use the graph of f to write an equation for each function whose graph is shown. (a) (b) (c) (d) (d) (c). Use the graph of f to write an equation for each function whose graph is shown. (a) (b) (c) In Eercises 8, identif the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. (d) (d)

80 In Eercises 9, g is related to one of the parent functions described in this chapter. (a) Identif the parent function f. (b) Describe the sequence of tranformations from f to g. (c) Sketch the graph of g. (d) Use function notation to write g in terms of f. 9. g 0. g 8. g 7. g. g. g 7. g. g 0 7. g 8. g 9. g 0. g 0. g. g. g 8. g 9. g. g 7. g 9 8. g 8 9. g 7 0. g. g. g In Eercises 0, write an equation for the function that is described b the given characteristics.. The shape of f, but moved two units to the right and eight units downward. The shape of f, but moved three units to the left, seven units upward, and reflected in the -ais. The shape of f, but moved units to the right. The shape of f, but moved si units to the left, si units downward, and reflected in the -ais 7. The shape of f, but moved 0 units upward and reflected in the -ais 8. The shape of f, but moved one unit to the left and seven units downward 9. The shape of f, but moved si units to the left and reflected in both the -ais and the -ais 0. The shape of f, but moved nine units downward and reflected in both the -ais and the -ais. Use the graph of f to write an equation for each function whose graph is shown. (a) (b). Use the graph of f to write an equation for each function whose graph is shown. (a) (b). Use the graph of to write an equation for each function whose graph is shown. (a) (b). Use the graph of f to write an equation for each function whose graph is shown. (a) (b) (, ) (, ) (, ) f (, ) 8 0 (, ) 8 (, 7) (, ) (, )

81 In Eercises 0, identif the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utilit to verif our answer Graphical Analsis In Eercises, use the viewing window shown to write a possible equation for the transformation of the parent function Graphical Reasoning In Eercises and, use the graph of f to sketch the graph of g... g (a) g f (c) g f (e) g f (a) g f (b) g f (c) g f (d) g f (e) g f (f) g f f 8 0 f Model It (b) g f (d) f (f) g f 7. Fuel Use The amounts of fuel F (in billions of gallons) used b trucks from 980 through 00 can be approimated b the function F f t t, 0 t where t represents the ear, with t 0 corresponding to 980. (Source: U.S. Federal Highwa Administration) (a) Describe the transformation of the parent function f. Then sketch the graph over the specified domain. (b) Find the average rate of change of the function from 980 to 00. Interpret our answer in the contet of the problem. (c) Rewrite the function so that t 0 represents 990. Eplain how ou got our answer. (d) Use the model from part (c) to predict the amount of fuel used b trucks in 00. Does our answer seem reasonable? Eplain.

82 8. Finance The amounts M (in trillions of dollars) of mortgage debt outstanding in the United States from 990 through 00 can be approimated b the function M f t 0.00 t 0.9, where t represents the ear, with t 0 corresponding to 990. (Source: Board of Governors of the Federal Reserve Sstem) (a) Describe the transformation of the parent function f. Then sketch the graph over the specified domain. (b) Rewrite the function so that t 0 represents 000. Eplain how ou got our answer. Snthesis True or False? In Eercises 9 and 70, determine whether the statement is true or false. Justif our answer. 9. The graphs of f and are identical. 70. If the graph of the parent function f is moved si units to the right, three units upward, and reflected in the -ais, then the point, 9 will lie on the graph of the transformation. 7. Describing Profits Management originall predicted that the profits from the sales of a new product would be approimated b the graph of the function f shown. The actual profits are shown b the function g along with a verbal description. Use the concepts of transformations of graphs to write g in terms of f. 0,000 0,000 (a) The profits were onl three-fourths as large as epected. (b) The profits were consistentl $0,000 greater than predicted. f f 0 t t 0,000 0,000 0,000 0,000 g g t t (c) There was a two-ear dela in the introduction of the product. After sales began, profits grew as epected. 7. Eplain wh the graph of f is a reflection of the graph of f about the -ais. 7. The graph of f passes through the points 0,,,, and,. Find the corresponding points on the graph of f. 7. Think About It You can use either of two methods to graph a function: plotting points or translating a parent function as shown in this section. Which method of graphing do ou prefer to use for each function? Eplain. (a) f (b) f Skills Review In Eercises 7 8, perform the operation and simplif ,000 0,000 In Eercises 8 and 8, evaluate the function at the specified values of the independent variable and simplif. 8. f (a) f (b) f (c) f 8. f 0 (a) f 0 (b) f (c) f 0 In Eercises 8 88, find the domain of the function. 8. f 8. f f f g t

83 Combinations of Functions: Composite Functions What ou should learn Add, subtract, multipl, and divide functions. Find the composition of one function with another function. Use combinations and compositions of functions to model and solve real-life problems. Wh ou should learn it Compositions of functions can be used to model and solve real-life problems. For instance, in Eercise 8, compositions of functions are used to determine the price of a new hbrid car. Arithmetic Combinations of Functions Just as two real numbers can be combined b the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be combined to create new functions. For eample, the functions given b f and g can be combined to form the sum, difference, product, and quotient of f and g. f g f g f g f g, ± Sum Difference Product Quotient The domain of an arithmetic combination of functions f and g consists of all real numbers that are common to the domains of f and g. In the case of the quotient f g, there is the further restriction that g 0.. Video Sum, Difference, Product, and Quotient of Functions Let f and g be two functions with overlapping domains. Then, for all common to both domains, the sum, difference, product, and quotient of f and g are defined as follows.. Sum: f g f g. Difference: f g f g. Product: fg f g f g f. Quotient: g 0 g, Eample Finding the Sum of Two Functions Given f and g, find f g. f g f g Now tr Eercise (a).

84 Eample Finding the Difference of Two Functions Given f and g, find f g. Then evaluate the difference when. The difference of f and g is f g f g. When, the value of this difference is f g. Now tr Eercise (b). In Eamples and, both f and g have domains that consist of all real numbers. So, the domains of f g and f g are also the set of all real numbers. Remember that an restrictions on the domains of f and g must be considered when forming the sum, difference, product, or quotient of f and g. Eample Finding the Domains of Quotients of Functions f g Find and for the functions given b f and g. Then find the domains of f g and g f. The quotient of f and g is g f f g and the quotient of g and f is g g f f g f. The domain of f is 0, and the domain of g is,. The intersection of these domains is 0,. So, the domains of f and g are as follows. g Domain of g f : 0, f g Domain of : 0, f Note that the domain of f g includes 0, but not, because ields a zero in the denominator, whereas the domain of g f includes, but not 0, because 0 ields a zero in the denominator. Now tr Eercise (d).

85 f g g() g f Domain of g Domain of f FIGURE 90 f(g()) Composition of Functions Another wa of combining two functions is to form the composition of one with the other. For instance, if f and g, the composition of f with g is f g f. This composition is denoted as f g and reads as f composed with g. Definition of Composition of Two Functions The composition of the function f with the function g is f g f g. The domain of f g is the set of all in the domain of g such that g is in the domain of f. (See Figure 90.). Video The following tables of values help illustrate the composition f g given in Eample. 0 g 0 g f g 0 0 f g Note that the first two tables can be combined (or composed ) to produce the values given in the third table. Eample Composition of Functions Given f and g, find the following. a. f g b. g f c. g f a. The composition of f with g is as follows. f g f g f b. The composition of g with f is as follows. g f g f g Definition of f g Definition of g Definition of f Simplif. Definition of g f Definition of f Definition of g Epand. Simplif. Note that, in this case, f g g f. c. Using the result of part (b), ou can write the following. g f Substitute. 8 Simplif. Simplif. Now tr Eercise.

86 You can use a graphing utilit to determine the domain of a composition of functions. For the composition in Eample, enter the function composition as You should obtain the graph shown below. Use the trace feature to determine that the -coordinates of points on the graph etend from to. So, the domain of f g is. Technolog Eample Finding the Domain of a Composite Function Given f 9 and g 9, find the composition f g. Then find the domain of f g. f g f g f From this, it might appear that the domain of the composition is the set of all real numbers. This, however is not true. Because the domain of f is the set of all real numbers and the domain of g is, the domain of f g is. Now tr Eercise. In Eamples and, ou formed the composition of two given functions. In calculus, it is also important to be able to identif two functions that make up a given composite function. For instance, the function h given b h is the composition of f with g, where f and g. That is, h g f g. Basicall, to decompose a composite function, look for an inner function and an outer function. In the function h above, g is the inner function and f is the outer function. Eample Decomposing a Composite Function Write the function given b h as a composition of two functions. One wa to write h as a composition of two functions is to take the inner function to be g and the outer function to be f. Then ou can write h f f g. Now tr Eercise 7.

87 Application Eample 7 Bacteria Count The number N of bacteria in a refrigerated food is given b N T 0T 80T 00, where T is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given b T t t, where t is the time in hours. (a) Find the composition N T t and interpret its meaning in contet. (b) Find the time when the bacterial count reaches 000. a. N T t 0 t 80 t 00 0 t t 0t t 0t 80 0t t 0 0 t T The composite function N T t represents the number of bacteria in the food as a function of the amount of time the food has been out of refrigeration. b. The bacterial count will reach 000 when 0t Solve this equation to find that the count will reach 000 when t. hours. When ou solve this equation, note that the negative value is rejected because it is not in the domain of the composite function. Now tr Eercise. W RITING ABOUT MATHEMATICS Analzing Arithmetic Combinations of Functions a. Use the graphs of f and f g in Figure 9 to make a table showing the values of g when,,,,, and. Eplain our reasoning. b. Use the graphs of f and f h in Figure 9 to make a table showing the values of h when,,,,, and. Eplain our reasoning. f f + g f h FIGURE 9

88 Eercises The smbol indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. Click on to view the complete solution of the eercise. Click on to print an enlarged cop of the graph. Click on to view the Make a Decision eercise. VOCABULARY CHECK: Fill in the blanks. Glossar. Two functions f and g can be combined b the arithmetic operations of,,, and to create new functions.. The of the function f with g is f g f g.. The domain of f g is all in the domain of g such that is in the domain of f.. To decompose a composite function, look for an function and an function. In Eercises, use the graphs of f and g to graph h f g..... In Eercises, find (a) f g, (b) f g, (c) fg, and (d) f /g. What is the domain of f /g?. f,. f, 7. f, 8. f, 9. f, 0... f g f, f, f f, g g g g g g g g g f g g f In Eercises, evaluate the indicated function for f and g.. f g. f g. f g 0. f g 7. f g t 8. f g t 9. fg 0. fg f. f. g g. g. fg f In Eercises 8, graph the functions f, g, and f g on the same set of coordinate aes f, f, f, f, g g g g Graphical Reasoning In Eercises 9 and 0, use a graphing utilit to graph f, g, and f g in the same viewing window. Which function contributes most to the magnitude of the sum when 0? Which function contributes most to the magnitude of the sum when >? f g f, f, In Eercises, find (a) f g, (b) g f, and (c) f f.. f,. f,. f, g 0 g g g g. f g,

89 In Eercises, find (a) f g and (b) g f. Find the domain of each function and each composite function f, f, f, f, f, In Eercises, use the graphs of f and g to evaluate the functions. = f(). (a) f g (b) f g. (a) f g (b) fg. (a) f g (b) g f. (a) f g (b) g f g g g f, g f, g f, g g g = g() In Eercises 7, find two functions f and g such that f g h. (There are man correct answers.) 7. h 8. h 9. h 0. h 9. h. h.. h 7 h 0 7. Stopping Distance The research and development department of an automobile manufacturer has determined that when a driver is required to stop quickl to avoid an accident, the distance (in feet) the car travels during the driver s reaction time is given b R, where is the speed of the car in miles per hour. The distance (in feet) traveled while the driver is braking is given b B. Find the function that represents the total stopping distance T. Graph the functions R, B, and T on the same set of coordinate aes for Sales From 000 to 00, the sales (in thousands of dollars) for one of two restaurants owned b the same parent compan can be modeled b R 80 8t 0.8t, where t 0 represents 000. During the same si-ear period, the sales R (in thousands of dollars) for the second restaurant can be modeled b R 0.78t, t 0,,,,,. (a) Write a function R that represents the total sales of the two restaurants owned b the same parent compan. (b) Use a graphing utilit to graph R, R, and R in the same viewing window. 7. Vital Statistics Let b t be the number of births in the United States in ear t, and let d t represent the number of deaths in the United States in ear t, where t 0 corresponds to 000. (a) If p t is the population of the United States in ear t, find the function c t that represents the percent change in the population of the United States. (b) Interpret the value of c. 8. Pets Let d t be the number of dogs in the United States in ear t, and let c t be the number of cats in the United States in ear t, where t 0 corresponds to 000. (a) Find the function p t that represents the total number of dogs and cats in the United States. (b) Interpret the value of p. (c) Let n t represent the population of the United States in ear t, where t 0 corresponds to 000. Find and interpret 9. Militar Personnel The total numbers of Arm personnel (in thousands) A and Nav personnel (in thousands) N from 990 to 00 can be approimated b the models A t.t 9.8t 7 and h t p t n t. N t.9t.t 0 t 0,,,,, where t represents the ear, with t 0 corresponding to 990. (Source: Department of Defense) (a) Find and interpret A N t. Evaluate this function for t, 8, and. (b) Find and interpret A N t. Evaluate this function for t, 8, and. R

90 0. Sales The sales of eercise equipment E (in millions of dollars) in the United States from 997 to 00 can be approimated b the function E t.9t.t and the U.S. population P (in millions) from 997 to 00 can be approimated b the function P t.0t.0 where t represents the ear, with t 7 corresponding to 997. (Source: National Sporting Goods Association, U.S. Census Bureau) (a) Find and interpret h t E t P t. (b) Evaluate the function in part (a) for t 7, 0, and.. Health Care Costs The table shows the total amounts (in billions of dollars) spent on health services and supplies in the United States (including Puerto Rico) for the ears 99 through 00. The variables,, and represent out-of-pocket paments, insurance premiums, and other tpes of paments, respectivel. (Source: Centers for Medicare and Medicaid Services) Year Model It. Graphical Reasoning An electronicall controlled thermostat in a home is programmed to lower the temperature automaticall during the night. The temperature in the house T (in degrees Fahrenheit) is given in terms of t, the time in hours on a -hour clock (see figure). Temperature (in F) T 9 8 Time (in hours) (a) Eplain wh T is a function of t. (b) Approimate T and T. (c) The thermostat is reprogrammed to produce a temperature H for which H t T t. How does this change the temperature? (d) The thermostat is reprogrammed to produce a temperature H for which H t T t. How does this change the temperature? (e) Write a piecewise-defined function that represents the graph.. Geometr A square concrete foundation is prepared as a base for a clindrical tank (see figure). t r (a) Use the regression feature of a graphing utilit to find a linear model for and quadratic models for and. Let t represent 99. (b) Find. What does this sum represent? (c) Use a graphing utilit to graph,,, and in the same viewing window. (d) Use the model from part (b) to estimate the total amounts spent on health services and supplies in the ears 008 and 00. (a) Write the radius r of the tank as a function of the length of the sides of the square. (b) Write the area A of the circular base of the tank as a function of the radius r. (c) Find and interpret A r.

91 . Phsics A pebble is dropped into a calm pond, causing ripples in the form of concentric circles (see figure). The radius r (in feet) of the outer ripple is r t 0.t, where t is the time in seconds after the pebble strikes the water. The area A of the circle is given b the function A r r. Find and interpret A r t. 8. Consumer Awareness The suggested retail price of a new hbrid car is p dollars. The dealership advertises a factor rebate of $000 and a 0% discount. (a) Write a function R in terms of p giving the cost of the hbrid car after receiving the rebate from the factor. (b) Write a function S in terms of p giving the cost of the hbrid car after receiving the dealership discount. (c) Form the composite functions R S p and S R p and interpret each. (d) Find R S 0,00 and S R 0,00. Which ields the lower cost for the hbrid car? Eplain. Snthesis. Bacteria Count The number N of bacteria in a refrigerated food is given b N T 0T 0T 00, where T is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given b T t t, where t is the time in hours. (a) Find the composition N T t and interpret its meaning in contet. (b) Find the time when the bacterial count reaches 00.. Cost The weekl cost C of producing units in a manufacturing process is given b C The number of units produced in t hours is given b t 0t. (a) Find and interpret C t. (b) Find the time that must elapse in order for the cost to increase to $, Salar You are a sales representative for a clothing manufacturer. You are paid an annual salar, plus a bonus of % of our sales over $00,000. Consider the two functions given b f 00,000 0 t and T 0 g() 0.0. If is greater than $00,000, which of the following represents our bonus? Eplain our reasoning. (a) f g (b) g f True or False? In Eercises 9 and 70, determine whether the statement is true or false. Justif our answer. 9. If f and g, then f g) g f ). 70. If ou are given two functions f and g, ou can calculate f g if and onl if the range of g is a subset of the domain of f. 7. Proof Prove that the product of two odd functions is an even function, and that the product of two even functions is an even function. 7. Conjecture Use eamples to hpothesize whether the product of an odd function and an even function is even or odd. Then prove our hpothesis. Skills Review Average Rate of Change difference quotient f h f h and simplif our answer. In Eercises 7 7, find the 7. f 7. f 7. f 7. f In Eercises 77 80, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line. 77.,, m 78.,, m 79. 8,, m 80. 7, 0, m 7

92 Inverse Functions What ou should learn Find inverse functions informall and verif that two functions are inverse functions of each other. Use graphs of functions to determine whether functions have inverse functions. Use the Horizontal Line Test to determine if functions are one-to-one. Find inverse functions algebraicall. Wh ou should learn it Inverse functions can be used to model and solve real-life problems. For instance, in Eercise 80, an inverse function can be used to determine the ear in which there was a given dollar amount of sales of digital cameras in the United States. Inverse Functions Recall from the Functions section, that a function can be represented b a set of ordered pairs. For instance, the function f from the set A,,, to the set B,, 7, 8 can be written as follows. f :,,,,, 7,, 8 In this case, b interchanging the first and second coordinates of each of these ordered pairs, ou can form the inverse function of f, which is denoted b f. It is a function from the set B to the set A, and can be written as follows. f :,,,, 7,, 8, Note that the domain of f is equal to the range of f, and vice versa, as shown in Figure 9. Also note that the functions f and f have the effect of undoing each other. In other words, when ou form the composition of f with f or the composition of f with f, ou obtain the identit function. f f f f f f f() = + Domain of f Range of f f() Range of f FIGURE 9 f () = Domain of f. Video Eample Finding Inverse Functions Informall Find the inverse function of f(). Then verif that both f f and f f are equal to the identit function. The function f multiplies each input b. To undo this function, ou need to divide each input b. So, the inverse function of f is f. You can verif that both f f and f f as follows. f f f Now tr Eercise. f f f

93 Eploration Consider the functions given b f and f. Evaluate f f and f f for the indicated values of. What can ou conclude about the functions? f f f f Definition of Inverse Function Let f and g be two functions such that f g for ever in the domain of g and g f for ever in the domain of f. Under these conditions, the function g is the inverse function of the function f. The function g is denoted b f (read f -inverse ). So, f f and f f. The domain of f must be equal to the range of f, and the range of f must be equal to the domain of f. Don t be confused b the use of to denote the inverse function f. In this tet, whenever f is written, it alwas refers to the inverse function of the function f and not to the reciprocal of f. If the function g is the inverse function of the function f, it must also be true that the function f is the inverse function of the function g. For this reason, ou can sa that the functions f and g are inverse functions of each other. Video Eample Verifing Inverse Functions. Which of the functions is the inverse function of g B forming the composition of f with g, ou have f g f. h Substitute for. Because this composition is not equal to the identit function, it follows that g is not the inverse function of f. B forming the composition of f with h, ou have f h f. f? So, it appears that h is the inverse function of f. You can confirm this b showing that the composition of h with f is also equal to the identit function. Now tr Eercise.

94 (a, b) FIGURE 9 (, ) (, 0) = f() f ( ) = ( + ) (b, a) = = f () (, ) (, ) = (0, ) FIGURE 9 (, ) f ( ) = (, ) (, ) (, ) The Graph of an Inverse Function The graphs of a function f and its inverse function f are related to each other in the following wa. If the point a, b lies on the graph of f, then the point b, a must lie on the graph of f, and vice versa. This means that the graph of f is a reflection of the graph of f in the line, as shown in Figure 9. Eample Finding Inverse Functions Graphicall Sketch the graphs of the inverse functions f and f on the same rectangular coordinate sstem and show that the graphs are reflections of each other in the line. The graphs of f and f are shown in Figure 9. It appears that the graphs are reflections of each other in the line. You can further verif this reflective propert b testing a few points on each graph. Note in the following list that if the point a, b is on the graph of f, the point b, a is on the graph of f. Graph of f Graph of f,, 0,, 0,,,,,, Now tr Eercise (, ) (, 9) f() = (, ) = (9, ) (, ) f () = (0, 0) FIGURE 9 Video Eample Finding Inverse Functions Graphicall Sketch the graphs of the inverse functions f 0 and f on the same rectangular coordinate sstem and show that the graphs are reflections of each other in the line. The graphs of f and f are shown in Figure 9. It appears that the graphs are reflections of each other in the line. You can further verif this reflective propert b testing a few points on each graph. Note in the following list that if the point a, b is on the graph of f, the point b, a is on the graph of f. Graph of f, 0 Graph of f 0, 0 0, 0,,,,, 9 9, Tr showing that f f and f f. Now tr Eercise 7.

95 One-to-One Functions The reflective propert of the graphs of inverse functions gives ou a nice geometric test for determining whether a function has an inverse function. This test is called the Horizontal Line Test for inverse functions. Horizontal Line Test for Inverse Functions A function f has an inverse function if and onl if no horizontal line intersects the graph of f at more than one point.. Video If no horizontal line intersects the graph of f at more than one point, then no -value is matched with more than one -value. This is the essential characteristic of what are called one-to-one functions. One-to-One Functions A function f is one-to-one if each value of the dependent variable corresponds to eactl one value of the independent variable. A function f has an inverse function if and onl if f is one-to-one. Consider the function given b f. The table on the left is a table of values for f. The table of values on the right is made up b interchanging the columns of the first table. The table on the right does not represent a function because the input is matched with two different outputs: and. So, f is not one-to-one and does not have an inverse function. f f() = FIGURE Eample Appling the Horizontal Line Test FIGURE 97 f() = a. The graph of the function given b f is shown in Figure 9. Because no horizontal line intersects the graph of f at more than one point, ou can conclude that f is a one-to-one function and does have an inverse function. b. The graph of the function given b f is shown in Figure 97. Because it is possible to find a horizontal line that intersects the graph of f at more than one point, ou can conclude that f is not a one-to-one function and does not have an inverse function. Now tr Eercise 9.

96 Note what happens when ou tr to find the inverse function of a function that is not one-to-one. f Original function Replace f() b. Interchange and. Isolate -term. ± Solve for. You obtain two -values for each. Finding Inverse Functions Algebraicall For simple functions (such as the one in Eample ), ou can find inverse functions b inspection. For more complicated functions, however, it is best to use the following guidelines. The ke step in these guidelines is Step interchanging the roles of and. This step corresponds to the fact that inverse functions have ordered pairs with the coordinates reversed. Finding an Inverse Function. Use the Horizontal Line Test to decide whether f has an inverse function.. In the equation for f, replace f b.. Interchange the roles of and, and solve for.. Replace b f in the new equation.. Verif that f and f are inverse functions of each other b showing that the domain of f is equal to the range of f, the range of f is equal to the domain of f, and f f and f f.. FIGURE 98 Video Eploration f () = Restrict the domain of f to 0. Use a graphing utilit to graph the function. Does the restricted function have an inverse function? Eplain. Eample Find the inverse function of Finding an Inverse Function Algebraicall f. The graph of f is a line, as shown in Figure 98. This graph passes the Horizontal Line Test. So, ou know that f is one-to-one and has an inverse function. f f f Write original function. Replace f b. Interchange and. Multipl each side b. Isolate the -term. Solve for. Replace b f. Note that both f and have domains and ranges that consist of the entire set of real numbers. Check that f f and f f. Now tr Eercise.

97 Eample 7 Finding an Inverse Function Find the inverse function of f. f ( ) = + The graph of f is a curve, as shown in Figure 99. Because this graph passes the Horizontal Line Test, ou know that f is one-to-one and has an inverse function. f Write original function. Replace f b. FIGURE 99 f Interchange and. Cube each side. Solve for. Replace b f. Both f and f have domains and ranges that consist of the entire set of real numbers. You can verif this result numericall as shown in the tables below. f f Now tr Eercise. W RITING ABOUT MATHEMATICS The Eistence of an Inverse Function Write a short paragraph describing wh the following functions do or do not have inverse functions. a. Let represent the retail price of an item (in dollars), and let f represent the sales ta on the item. Assume that the sales ta is % of the retail price and that the sales ta is rounded to the nearest cent. Does this function have an inverse function? (Hint: Can ou undo this function? For instance, if ou know that the sales ta is $0., can ou determine eactl what the retail price is?) b. Let represent the temperature in degrees Celsius, and let f represent the temperature in degrees Fahrenheit. Does this function have an inverse function? (Hint: The formula for converting from degrees Celsius to degrees Fahrenheit is F 9 ) C.

98 Eercises The smbol Click on Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarged cop of the graph. to view the Make a Decision eercise. Glossar VOCABULARY CHECK: Fill in the blanks.. If the composite functions f g and g f then the function g is the function of f.. The domain of f is the of f, and the of f is the range of f.. The graphs of f and f are reflections of each other in the line.. A function f is if each value of the dependent variable corresponds to eactl one value of the independent variable.. A graphical test for the eistence of an inverse function of f is called the Line Test. In Eercises 8, find the inverse function of f informall. Verif that and f f f f.. f. f. f 9. f. f. f 7. f 8. f.. In Eercises 9, match the graph of the function with the graph of its inverse function. [The graphs of the inverse functions are labeled (a), (b), (c), and (d).] In Eercises, show that f and g are inverse functions (a) algebraicall and (b) graphicall. (a) (c) (b) (d) f, 0. f,. f 9,.. f, f, f 7, f, f 8, f, f, f, 0, 0, g g g 7 g g 8 g g,. f g, g, g g 9, g <

99 In Eercises and, does the function have an inverse function?.. In Eercises 7 and 8, use the table of values for f to complete a table for f f 0 f f 0 0 f In Eercises 9, (a) find the inverse function of f, (b) graph both f and f on the same set of coordinate aes, (c) describe the relationship between the graphs of f and f, and (d) state the domain and range of f and f. 9. f 0. f. f. f. f. f, 0.. f, f, f 8. f 9. f 0. f. f. f 8. f. f In Eercises 8, determine whether the function has an inverse function. If it does, find the inverse function. In Eercises 9, does the function have an inverse function? f. 7. g p 0.. f,.. f, < 0, 0.. h. f f f q f,, f, 0 > 0 In Eercises 8, use a graphing utilit to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. 7. f 8. f. g h g f f 8 f 0

100 In Eercises 9 7, use the functions given b f 8 and g to find the indicated value or function. 9. f g 70. g f 7. f f 7. g g 7. ( f g) 7. g f In Eercises 7 78, use the functions given b f and g to find the specified function. 7. g f 7. f g 77. f g 78. g f 79. U.S. Households The numbers of households f (in thousands) in the United States from 99 to 00 are shown in the table. The time (in ears) is given b t, with t corresponding to 99. (Source: U.S. Census Bureau) Year, t Model It Households, f t 98,990 99,7 7 0,08 8 0,8 9 0,87 0 0,70 08,09 09,97,78 (a) Find f 08,09. (b) What does f mean in the contet of the problem? (c) Use the regression feature of a graphing utilit to find a linear model for the data, m b. (Round m and b to two decimal places.) (d) Algebraicall find the inverse function of the linear model in part (c). (e) Use the inverse function of the linear model ou found in part (d) to approimate f 7, 0. (f) Use the inverse function of the linear model ou found in part (d) to approimate f 08,09. How does this value compare with the original data shown in the table? 80. Digital Camera Sales The factor sales f (in millions of dollars) of digital cameras in the United States from 998 through 00 are shown in the table. The time (in ears) is given b t, with t 8 corresponding to 998. (Source: Consumer Electronincs Association) Year, t Sales, f t (a) Does f eist? (b) If f eists, what does it represent in the contet of the problem? (c) If f eists, find f 8. (d) If the table was etended to 00 and if the factor sales of digital cameras for that ear was $79 million, would f eist? Eplain. 8. Miles Traveled The total numbers f (in billions) of miles traveled b motor vehicles in the United States from 99 through 00 are shown in the table. The time (in ears) is given b t, with t corresponding to 99. (Source: U.S. Federal Highwa Administration) Year, t Miles traveled, f t (a) Does f eist? (b) If f eists, what does it mean in the contet of the problem? (c) If f eists, find f. (d) If the table was etended to 00 and if the total number of miles traveled b motor vehicles for that ear was 77 billion, would f eist? Eplain.

101 8. Hourl Wage Your wage is $8.00 per hour plus $0.7 for each unit produced per hour. So, our hourl wage in terms of the number of units produced is (a) Find the inverse function. (b) What does each variable represent in the inverse function? (c) Determine the number of units produced when our hourl wage is $.. 8. Diesel Mechanics The function given b 0.0.0, approimates the ehaust temperature in degrees Fahrenheit, where is the percent load for a diesel engine. (a) Find the inverse function. What does each variable represent in the inverse function? (b) Use a graphing utilit to graph the inverse function. (c) The ehaust temperature of the engine must not eceed 00 degrees Fahrenheit. What is the percent load interval? 8. Cost You need a total of 0 pounds of two tpes of ground beef costing $. and $.0 per pound, respectivel. A model for the total cost of the two tpes of beef is..0 0 where is the number of pounds of the less epensive ground beef. (a) Find the inverse function of the cost function. What does each variable represent in the inverse function? (b) Use the contet of the problem to determine the domain of the inverse function. (c) Determine the number of pounds of the less epensive ground beef purchased when the total cost is $7. Snthesis 0 < < 00 True or False? In Eercises 8 and 8, determine whether the statement is true or false. Justif our answer. f 8. If f is an even function, eists. 8. If the inverse function of f eists and the graph of f has a -intercept, the -intercept of f is an -intercept of f. 87. Proof Prove that if f and g are one-to-one functions, then f g g f. 88. Proof Prove that if f is a one-to-one odd function, then f is an odd function. In Eercises 89 9, use the graph of the function f to create a table of values for the given points. Then create a second table that can be used to find f, and sketch the graph of if possible. f f 9. Think About It The function given b f k has an inverse function, and f (). Find k. 9. Think About It The function given b f k has an inverse function, and f ( ). Find k. Skills Review 8 In Eercises 9 0, solve the equation using an convenient method f 0. Find two consecutive positive even integers whose product is Geometr A triangular sign has a height that is twice its base. The area of the sign is 0 square feet. Find the base and height of the sign. f f

102 Mathematical Modeling and Variation.. Insured commercial banks (in thousands) What ou should learn Use mathematical models to approimate sets of data points. Use the regression feature of a graphing utilit to find the equation of a least squares regression line. Write mathematical models for direct variation. Write mathematical models for direct variation as an nth power. Write mathematical models for inverse variation. Write mathematical models for joint variation. Wh ou should learn it You can use functions as models to represent a wide variet of real-life data sets. For instance, in Eercise 7, a variation model can be used to model the water temperature of the ocean at various depths. Video Video FIGURE 00 U.S. Banks = 0.8t Year ( 99) t Introduction You have alread studied some techniques for fitting models to data. For instance, in the Linear Equations in Two Variables section, ou learned how to find the equation of a line that passes through two points. In this section, ou will stud other techniques for fitting models to data: least squares regression and direct and inverse variation. The resulting models are either polnomial functions or rational functions. (Rational functions will be studied in the net chapter.) Eample A Mathematical Model The numbers of insured commercial banks (in thousands) in the United States for the ears 99 to 00 are shown in the table. (Source: Federal Deposit Insurance Corporation) Year A linear model that approimates the data is 0.8t. for t, where t is the ear, with t corresponding to 99. Plot the actual data and the model on the same graph. How closel does the model represent the data? The actual data are plotted in Figure 00, along with the graph of the linear model. From the graph, it appears that the model is a good fit for the actual data. You can see how well the model fits b comparing the actual values of with the values of given b the model. The values given b the model are labeled * in the table below. t * Now tr Eercise. Insured commercial banks, Note in Eample that ou could have chosen an two points to find a line that fits the data. However, the given linear model was found using the regression feature of a graphing utilit and is the line that best fits the data. This concept of a best-fitting line is discussed on the net page.

103 .. Prize mone (in millions of dollars) Video Simulation FIGURE 0 p Indianapolis Year ( 99) t p p* t Least Squares Regression and Graphing Utilities So far in this tet, ou have worked with man different tpes of mathematical models that approimate real-life data. In some instances the model was given (as in Eample ), whereas in other instances ou were asked to find the model using simple algebraic techniques or a graphing utilit. To find a model that approimates the data most accuratel, statisticians use a measure called the sum of square differences, which is the sum of the squares of the differences between actual data values and model values. The bestfitting linear model, called the least squares regression line, is the one with the least sum of square differences. Recall that ou can approimate this line visuall b plotting the data points and drawing the line that appears to fit best or ou can enter the data points into a calculator or computer and use the linear regression feature of the calculator or computer. When ou use the regression feauture of a graphing calculator or computer program, ou will notice that the program ma also output an r-value. This r-value is the correlation coefficient of the data and gives a measure of how well the model fits the data. The closer the value of is to, the better the fit. r Eample Finding a Least Squares Regression Line The amounts p (in millions of dollars) of total annual prize mone awarded at the Indianapolis 00 race from 99 to 00 are shown in the table. Construct a scatter plot that represents the data and find the least squares regression line for the data. (Source: ind00.com) Let t represent 99. The scatter plot for the points is shown in Figure 0. Using the regression feature of a graphing utilit, ou can determine that the equation of the least squares regression line is p 0.8t.. Year Prize mone, p To check this model, compare the actual p-values with the p-values given b the model, which are labeled p* in the table at the left. The correlation coefficient for this model is r 0.99, which implies that the model is a good fit. Now tr Eercise 7.

104 Direct Variation There are two basic tpes of linear models. The more general model has a -intercept that is nonzero. m b, b 0 The simpler model k has a -intercept that is zero. In the simpler model, is said to var directl as, or to be directl proportional to. Direct Variation The following statements are equivalent.. varies directl as.. is directl proportional to.. k for some nonzero constant k. k is the constant of variation or the constant of proportionalit. Video Eample Direct Variation. Video. Pennslvania Taes 00 = (00,.0) Gross income (in dollars) State income ta (in dollars) FIGURE 0 In Pennslvania, the state income ta is directl proportional to gross income. You are working in Pennslvania and our state income ta deduction is $.0 for a gross monthl income of $00. Find a mathematical model that gives the Pennslvania state income ta in terms of gross income. Verbal Model: State income ta k Gross income Labels: (dollars) (dollars) (percent in decimal form) Equation: k To solve for k, substitute the given information into the equation k, and then solve for k. k Write direct variation model..0 k 00 Substitute.0 and k Simplif. So, the equation (or model) for state income ta in Pennslvania is State income ta Gross income Income ta rate k In other words, Pennslvania has a state income ta rate of.07% of gross income. The graph of this equation is shown in Figure 0. Now tr Eercise.

105 Direct Variation as an nth Power Another tpe of direct variation relates one variable to a power of another variable. For eample, in the formula for the area of a circle A r the area A is directl proportional to the square of the radius r. Note that for this formula, is the constant of proportionalit. Note that the direct variation model k is a special case of k n with n. Direct Variation as an nth Power The following statements are equivalent.. varies directl as the nth power of.. is directl proportional to the nth power of.. k n for some constant k. Eample Direct Variation as nth Power t = 0 sec t = sec FIGURE t = sec The distance a ball rolls down an inclined plane is directl proportional to the square of the time it rolls. During the first second, the ball rolls 8 feet. (See Figure 0.) a. Write an equation relating the distance traveled to the time. b. How far will the ball roll during the first seconds? a. Letting d be the distance (in feet) the ball rolls and letting t be the time (in seconds), ou have d kt. Now, because d 8 when t, ou can see that k 8, as follows. d kt 8 k 8 k So, the equation relating distance to time is d 8t. b. When t, the distance traveled is d feet. Now tr Eercise. In Eamples and, the direct variations are such that an increase in one variable corresponds to an increase in the other variable. This is also true in the model d F, F > 0, where an increase in F results in an increase in d. You should not, however, assume that this alwas occurs with direct variation. For eample, in the model, an increase in results in a decrease in, and et is said to var directl as.

106 Inverse Variation. Video Inverse Variation The following statements are equivalent.. varies inversel as.. is inversel proportional to.. k for some constant k. If and are related b an equation of the form k n, then varies inversel as the nth power of (or is inversel proportional to the nth power of ). Some applications of variation involve problems with both direct and inverse variation in the same model. These tpes of models are said to have combined variation. Video Eample Direct and Inverse Variation. P V V P > P then V < V P FIGURE 0 If the temperature is held constant and pressure increases, volume decreases. A gas law states that the volume of an enclosed gas varies directl as the temperature and inversel as the pressure, as shown in Figure 0. The pressure of a gas is 0.7 kilogram per square centimeter when the temperature is 9 K and the volume is 8000 cubic centimeters. (a) Write an equation relating pressure, temperature, and volume. (b) Find the pressure when the temperature is 00 K and the volume is 7000 cubic centimeters. a. Let V be volume (in cubic centimeters), let P be pressure (in kilograms per square centimeter), and let T be temperature (in Kelvin). Because V varies directl as T and inversel as P, ou have V kt P. Now, because P 0.7 when T 9 and V 8000, ou have 8000 k k So, the equation relating pressure, temperature, and volume is V T P b. When T 00 and V 7000, the pressure is P 000 kilogram per square centimeter Now tr Eercise.

107 Video Joint Variation In Eample, note that when a direct variation and an inverse variation occur in the same statement, the are coupled with the word and. To describe two different direct variations in the same statement, the word jointl is used.. Joint Variation The following statements are equivalent.. z varies jointl as and.. z is jointl proportional to and.. z k for some constant k. If,, and z are related b an equation of the form z k n m then z varies jointl as the nth power of and the mth power of. Video Eample Joint Variation. The simple interest for a certain savings account is jointl proportional to the time and the principal. After one quarter ( months), the interest on a principal of $000 is $.7. a. Write an equation relating the interest, principal, and time. b. Find the interest after three quarters. a. Let I interest (in dollars), P principal (in dollars), and t time (in ears). Because I is jointl proportional to P and t, ou have I kpt. For I.7, P 000, and t, ou have.7 k 000 which implies that k So, the equation relating interest, principal, and time is I 0.0Pt which is the familiar equation for simple interest where the constant of proportionalit, 0.0, represents an annual interest rate of.%. b. When P $000 and t, the interest is I $.. Now tr Eercise 7.

108 Eercises The smbol indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. Click on to view the complete solution of the eercise. Click on to print an enlarged cop of the graph. Click on to view the Make a Decision eercise. Glossar VOCABULARY CHECK: Fill in the blanks.. Two techniques for fitting models to data are called direct and least squares.. Statisticians use a measure called of to find a model that approimates a set of data most accuratel.. An r-value of a set of data, also called a, gives a measure of how well a model fits a set of data.. Direct variation models can be described as varies directl as, or is to.. In direct variation models of the form k, k is called the of.. The direct variation model k n can be described as varies directl as the nth power of, or is to the nth power of. 7. The mathematical model k is an eample of variation. 8. Mathematical models that involve both direct and inverse variation are said to have variation. 9. The joint variation model z k can be described as z varies jointl as and, or z is to and.. Emploment The total numbers of emploees (in thousands) in the United States from 99 to 00 are given b the following ordered pairs. 99, 8,0 99, 9,00 99,,0 99,,0 99,,9 997,,97 A linear model that approimates the data is 77.0t,9, where represents the number of emploees (in thousands) and t represents 99. Plot the actual data and the model on the same set of coordinate aes. How closel does the model represent the data? (Source: U.S. Bureau of Labor Statistics). Sports The winning times (in minutes) in the women s 00-meter freestle swimming event in the Olmpics from 98 to 00 are given b the following ordered pairs. 98,.0 9,.0 9,.9 90,.8 9,.7 98,. 97,. 97,. 998, 7,7 999, 9,8 000,,8 00,,7 00,,8 980,. 98,. 988,.0 99,. 99,. 000,.0 00,.09 A linear model that approimates the data is 0.0t.0, where represents the winning time (in minutes) and t 0 represents 90. Plot the actual data and the model on the same set of coordinate aes. How closel does the model represent the data? Does it appear that another tpe of model ma be a better fit? Eplain. (Source: The World Almanac and Book of Facts) In Eercises, sketch the line that ou think best approimates the data in the scatter plot.then find an equation of the line.....

109 7. Sports The lengths (in feet) of the winning men s discus throws in the Olmpics from 9 to 00 are listed below. (Source: The World Almanac and Book of Facts) (a) Sketch a scatter plot of the data. Let represent the length of the winning discus throw (in feet) and let t represent 9. (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utilit to find the least squares regression line that fits the data. (d) Compare the linear model ou found in part (b) with the linear model given b the graphing utilit in part (c). (e) Use the models from parts (b) and (c) to estimate the winning men s discus throw in the ear 008. (f) Use our school s librar, the Internet, or some other reference source to analze the accurac of the estimate in part (e). 8. Revenue The total revenues (in millions of dollars) for Outback Steakhouse from 99 to 00 are listed below. (Source: Outback Steakhouse, Inc.) (a) Sketch a scatter plot of the data. Let represent the total revenue (in millions of dollars) and let t represent 99. (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utilit to find the least squares regression line that fits the data. (d) Compare the linear model ou found in part (b) with the linear model given b the graphing utilit in part (c). (e) Use the models from parts (b) and (c) to estimate the revenues of Outback Steakhouse in 00. (f) Use our school s librar, the Internet, or some other reference source to analze the accurac of the estimate in part (e). 9. Data Analsis: Broadwa Shows The table shows the annual gross ticket sales S (in millions of dollars) for Broadwa shows in New York Cit from 99 through 00. (Source: The League of American Theatres and Producers, Inc.) Year Sales, S (a) Use a graphing utilit to create a scatter plot of the data. Let t represent 99. (b) Use the regression feature of a graphing utilit to find the equation of the least squares regression line that fits the data. (c) Use the graphing utilit to graph the scatter plot ou found in part (a) and the model ou found in part (b) in the same viewing window. How closel does the model represent the data? (d) Use the model to estimate the annual gross ticket sales in 00 and 007. (e) Interpret the meaning of the slope of the linear model in the contet of the problem. 0. Data Analsis: Television Households The table shows the numbers (in millions) of households with cable television and the numbers (in millions) of households with color television sets in the United States from 99 through 00. (Source: Nielson Media Research; Television Bureau of Advertising, Inc.) Households with cable, Households with color TV,