1.2. Graphs of Equations. The Graph of an Equation. What you should learn. Why you should learn it

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1 3330_010.qd 1 1/7/05 Chapter :31 AM Page 1 Function and Their Graphs 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 75 on page, a graph can be used to estimate the life epectancies of children who are born in the ears 005 and 010. The Graph of an Equation In Section 1.1, 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 3 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, 1, is a solution of 7 3 because 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 1 Determining s Determine whether (a), 13 and (b) 1, 3 are solutions of the equation a. 10 7? Write original equation. Substitute for and 13 for., 13 is a solution. Because the substitution does satisf the original equation, ou can conclude that the ordered pair, 13 is a solution of the original equation. b Write original equation.? Substitute 1 for and 3 for , 3 is not a solution. Because the substitution does not satisf the original equation, ou can conclude that the ordered pair 1, 3 is not a solution of the original equation. Now tr Eercise 1. John Griffin/The Image Works The basic technique used for sketching the graph of an equation is the point-plotting method. Sketching the Graph of an Equation b Point Plotting 1. 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. 3. Plot these points on a rectangular coordinate sstem.. Connect the points with a smooth curve or line.

2 3330_010.qd 1/7/05 8:31 AM Page 15 Section 1. Graphs of Equations 15 Eample Sketching the Graph of an Equation Sketch the graph of 7 3. Because the equation is alread solved for, construct a table of values that consists of several solution points of the equation. For instance, when 1, which implies that 1, 10 is a solution point of the graph. 7 3, , , 7 1 1, 1, 1 3 3, 5, 5 From the table, it follows that 1, 10, 0, 7, 1,,, 1, 3,, and, 5 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. ( 1, 10) 8 (0, 7) (1, ) (, 1) 8 10 (3, ) (, 5) FIGURE 1.15 Now tr Eercise 5.

3 3330_010.qd 1/7/05 8:31 AM Page 1 1 Chapter 1 Function and Their Graphs Eample 3 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 3 has the form a b c and its graph is a parabola ,, 1, 1 0, 1, 1, 3, 7 Net, plot the points given in the table, as shown in Figure 1.1. Finall, connect the points with a smooth curve, as shown in Figure (, ) ( 1, 1) (, ) (3, 7) (1, 1) (0, ) (, ) ( 1, 1) (, ) (3, 7) (1, 1) (0, ) = FIGURE 1.1 FIGURE 1.17 Now tr Eercise 7. The point-plotting method demonstrated in Eamples and 3 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,, 1, 1, 1, 1, and, in Figure 1.1 were plotted, an one of the three graphs in Figure 1.18 would be reasonable. FIGURE 1.18

4 3330_010.qd 1/7/05 8:31 AM Page 17 Section 1. Graphs of Equations 17 No -intercepts; one -intercept Technolog To graph an equation involving and on a graphing utilit, use the following procedure. 1. Rewrite the equation so that is isolated on the left side.. Enter the equation into the graphing utilit. 3. 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 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 1. To find -intercepts, let be zero and solve the equation for.. To find -intercepts, let be zero and solve the equation for. No intercepts FIGURE 1.19 = 3 (, 0) FIGURE 1.0 (0, 0) (, 0) Eample Finding - and -Intercepts Find the - and -intercepts of the graph of 3. Let 0. Then 0 3 has solutions 0 and ±. -intercepts: 0, 0,, 0,, 0 Let 0. Then has one solution, 0. -intercept: 0, 0 See Figure 1.0. Now tr Eercise 11.

5 3330_010.qd 1/7/05 8:31 AM Page Chapter 1 Function and Their Graphs 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 1.1. (, ) (, ) (, ) (, ) (, ) (, ) -ais smmetr -ais smmetr Origin smmetr FIGURE 1.1 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. ( 3, 7) 7 (3, 7) 5 3 (, ) (, ) ( 1, 1) (1, 1) FIGURE 1. 3 = -ais smmetr Graphical Tests for Smmetr 1. 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. 3. A graph is smmetric with respect to the origin if, whenever, is on the graph,, is also on the graph. Eample 5 Testing for Smmetr The graph of is smmetric with respect to the -ais because the point, is also on the graph of. (See Figure 1..) The table below confirms that the graph is smmetric with respect to the -ais , 3, 7, 1, 1 1, 1, 3, 7 Now tr Eercise 3.

6 3330_010.qd 1/7/05 8:31 AM Page 19 Section 1. Graphs of Equations 19 Algebraic Tests for Smmetr 1. 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. 3. The graph of an equation is smmetric with respect to the origin if replacing with and with ields an equivalent equation. 1 1 (1, 0) FIGURE 1.3 = 1 (, 1) (5, ) 3 5 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 1. Of the three tests for smmetr, the onl one that is satisfied is the test for -ais smmetr because 1 is equivalent to 1. 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 1 1, 0 1, 1 5 5, Now tr Eercise 37. Eample 7 Sketching the Graph of an Equation 3 5 (, 3) 1 = 1 3 (, 3) ( 1, ) (3, ) (0, 1) (, 1) FIGURE 1. (1, 0) 3 5 Sketch the graph of 1. 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 1.. From the table, ou can see that 0 when 1. So, the -intercept is 0, 1. Similarl, 0 when 1. So, the -intercept is 1, ,, 3 1, 0, 1 1, 0, 1 3,, 3 Now tr Eercise 1.

7 3330_010.qd 1/7/05 8:31 AM Page 0 0 Chapter 1 Function and Their Graphs FIGURE 1.5 Center: (h, k) Radius: r Point on circle: (, ) 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 a b c is a parabola (see Eample 3). The graph of a circle is also eas to recognize. Circles Consider the circle shown in Figure 1.5. 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. 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 1 and, using subtraction. FIGURE , So, h 1and k. ( 1, ) (3, ) 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 Eample 8 Finding the Equation of a Circle The point 3, lies on a circle whose center is at 1,, as shown in Figure 1.. Write the standard form of the equation of this circle. The radius of the circle is the distance between 1, and 3,. r h k Distance Formula 3 1 Substitute for,, h, and k. Simplif. 1 Simplif. 0 Radius Using h, k 1, and r 0, the equation of the circle is h k r Equation of circle 1 0 Substitute for h, k, and r Standard form Now tr Eercise 1.

8 3330_010.qd 1/7/05 8:31 AM Page 1 Section 1. Graphs of Equations 1 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. Additional Eamples a. Write the standard form of the equation of a circle whose endpoints of a diameter are, 1 and, 7. Answer: b. Identif an smmetr:. Answer: -ais smmetr 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 5 to 59 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 71 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 1.7. From the graph, ou can estimate that a height of 71 inches corresponds to a weight of about 11 pounds. Weight (in pounds) FIGURE Recommended Weight Height (in inches) c. To confirm algebraicall the estimate found in part (b), ou can substitute 71 for in the model (71).99(71) So, the graphical estimate of 11 pounds is fairl good. Now tr Eercise 75.

9 3330_010.qd 1/7/05 8:31 AM Page Chapter 1 Function and Their Graphs 1. Eercises VOCABULARY CHECK: Fill in the blanks. 1. 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. 3. 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. 5. 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. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Eercises 1, determine whether each point lies on the graph of the equation. Equation Points 1. (a) 0, (b) 5, 3. 3 (a), 0 (b), 8 3. (a) 1, 5 (b), (a), 1 (b) 3, 9 In Eercises 5 8, complete the table. Use the resulting solution points to sketch the graph of the equation , , , , In Eercises 9 0, find the - and -intercepts of the graph of the equation

10 3330_010.qd 1/7/05 8:31 AM Page 3 Section 1. Graphs of Equations 3 In Eercises 1, assume that the graph has the indicated tpe of smmetr. Sketch the complete graph of the equation. To print an enlarged cop of the graph, go to the website ais smmetr 3.. Origin smmetr 8 -ais smmetr -ais smmetr In Eercises 57, write the standard form of the equation of the circle with the given characteristics. 57. Center: 0, 0 ; radius: 58. Center: 0, 0 ; radius: Center:, 1 ; radius: 0. Center: 7, ; radius: 7 1. Center: 1, ; solution point: 0, 0. Center: 3, ; solution point: 1, 1 3. Endpoints of a diameter: 0, 0,, 8. Endpoints of a diameter:, 1,, 1 In Eercises 5 70, find the center and radius of the circle, and sketch its graph In Eercises 5 3, use the algebraic tests to check for smmetr with respect to both aes and the origin In Eercises 33, use smmetr to sketch the graph of the equation In Eercises 5 5, use a graphing utilit to graph the equation. Use a standard setting. Approimate an intercepts The smbol 71. Depreciation A manufacturing plant purchases a new molding machine for $5,000. The depreciated value (reduced value) after t ears is given b 5,000 0,000t, 0 t 8. Sketch the graph of the equation. 7. Consumerism You purchase a jet ski for $8100. The depreciated value after t ears is given b t, 0 t. Sketch the graph of the equation. 73. Geometr A regulation NFL plaing field (including the end zones) of length and width has a perimeter of or ards. 3 3 (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 and its area is A indicates an eercise or a part of an eercise in which ou are instructed to use a graphing utilit (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).

11 3330_010.qd 1/7/05 8:31 AM Page Chapter 1 Function and Their Graphs 7. Geometr A soccer plaing field of length and width has a perimeter of 30 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 180 and its area is A 180. (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 75. Population Statistics The table shows the life epectancies of a child (at birth) in the United States for selected ears from 190 to 000. (Source: U.S. National Center for Health Statistics) 7. Electronics The resistance (in ohms) of 1000 feet of solid copper wire at 8 degrees Fahrenheit can be approimated b the model 10, , where is the diameter of the wire in mils (0.001 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 (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 0.005t 0.57t.5, 0 t 100 where represents the life epectanc and t is the time in ears, with t 0 corresponding to 190. (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 198 both graphicall and algebraicall. (d) Use the graph of the model to estimate the life epectancies of a child for the ears 005 and 010. (e) Do ou think this model can be used to predict the life epectanc of a child 50 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 3 is smmetric with respect to (a) the -ais and (b) the origin. (There are man correct answers.) Skills Review 81. Identif the terms: Rewrite the epression using eponential notation. ( ) In Eercises 83 88, simplif the epression t 88. 3