1.1. Rectangular Coordinates. The Cartesian Plane. What you should learn. Why you should learn it. Plotting Points in the Cartesian Plane.

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1 0_00.qd /7/0 Chapter. 8:9 AM Page Functions and Their Graphs Rectangular Coordinates What ou should learn The Cartesian Plane 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. 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. -ais Wh ou should learn it The Cartesian plane can be used to represent relationships between two variables. For instance, in Eercise 0 on page, a graph represents the minimum wage in the United States from 90 to 00. Quadrant II Origin Quadrant I Directed distance (Vertical number line) -ais Quadrant III FIGURE -ais (, ) (Horizontal number line) Directed distance Quadrant IV. FIGURE -ais. 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. Ariel Skell/Corbis Eample (, ) Plotting Points in the Cartesian Plane (, ) Plot the points (, ), (, ), (0, 0), (, 0), and (, ). (, ) FIGURE. (0, 0) (, 0) 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 0_00.qd /7/0 8:9 AM Page Section. Rectangular Coordinates 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, 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 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). The HM mathspace CD-ROM and Eduspace for this tet contain additional resources related to the concepts discussed in this chapter. 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 0_00.qd /7/0 8:9 AM Page Chapter Functions and Their Graphs 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. 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.. FIGURE. (, ) (, ) The Distance Formula The distance d between the points, and, in the plane is d. 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 0_00.qd /7/0 8:9 AM Page 7 (, 7) d = d = 0 d = (, ) (, 0) 7 FIGURE.8 Eample Verifing a Right Triangle Section. Rectangular Coordinates 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. An overhead projector is useful for showing how to plot points and equations. Tr projecting a grid onto the chalkboard and then plotting points on the chalkboard, or tr using overhead markers and graph directl on the transparenc. A viewscreen, a device used with an overhead projector to project a graphing calculator s screen image, is also useful. Eercises on page 0 help develop a general understanding of the Midpoint Formula. 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,. For a proof of the Midpoint Formula, see Proofs in Mathematics on page. 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 0_00.qd /7/0 8:9 AM Page Chapter Functions and Their Graphs Applications Eample Finding the Length of a Pass Distance (in ards) Football Pass 0 (0, 8) 0 0 (0, ) Distance (in ards) FIGURE.0 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.) 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 0_00.qd /7/0 8:0 AM Page 7 Section. Eample 8 7 Rectangular Coordinates 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.. Paul Morrell (, ) (, ) Much of computer graphics, including this computer-generated goldfish tessellation, 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. Activities. Set up a Cartesian plane and plot the points, 0 and,.. Find the distance between, and,. Answer:. Find the midpoint of the line segment joining the points, and,. Answer:, 7 (, ) FIGURE 7. 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 V lwh V r h V r P l w C r A bh P a b c w r l h c a h w b r r l h

7 0_00.qd /7/0 8:0 AM Page 8 8 Chapter Functions and Their Graphs 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. Alternative Writing About Mathematics Use our school s librar, the Internet, or some other reference source to locate a set of real data that can be considered as ordered pairs. Decide which variables should be the - and -variables, and eplain wh ou made this decision. List each ordered pair. Sketch a scatter plot of the data, choosing appropriate aes and scale. Be sure to label each ais and give the units for each variable if necessar. Describe an apparent trends in the data. What can ou learn from our graph? 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 0_00.qd /7/0 8:0 AM Page 9 Section. Rectangular Coordinates 9. Eercises The HM mathspace CD-ROM and Eduspace for this tet contain step-b-step solutions to all odd-numbered eercises. The also provide Tutorial Eercises for additional help. 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.. 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. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at 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.), Number of stores,

9 0_00.qd /7/0 8:0 AM Page 0 0 Chapter Functions and Their Graphs. 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.).,,,.,, 8,.,,,.,,, 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, (, ) (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 0_00.qd /7/0 8:0 AM Page Section. Rectangular Coordinates 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... (, ) (, ) units (, ) units Sales (in millions) 00 $ 00 $7 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 Average price (in dollars per pound) (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?

11 0_00.qd /7/0 8:0 AM Page Chapter Functions and Their Graphs 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 (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), 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 0_00.qd /7/0 8:0 AM Page Section. Rectangular Coordinates 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), 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) 0, 0 Skills Review ( b, c ) ( a + b, c) (d) 77. 0, , , 0 In Eercises 8 88, solve the equation or inequalit < ( a, 0) < 88. (, ) 0 0