2 variables. is the same value as the solution of. 1 variable. You can use similar reasoning to solve quadratic equations. Work with a partner.

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1 9. b Graphing Essential Question How can ou use a graph to solve a quadratic equation in one variable? Based on what ou learned about the -intercepts of a graph in Section., it follows that the -intercept of the graph of the linear equation = a + b variables The -intercept of the graph of = + is. a + b = 0. variable The solution of the equation + = 0 is =. is the same value as the solution of You can use similar reasoning to solve quadratic equations. (, 0) Solving a Quadratic Equation b Graphing Work with a partner. 0 a. Sketch the graph of =. 8 b. What is the definition of an -intercept of a graph? How man -intercepts does this graph have? What are the? c. What is the definition of a solution of an equation in? How man solutions does the equation = 0 have? What are the? d. Eplain how ou can verif the solutions ou found in part (c). b Graphing MAKING SENSE OF PROBLEMS To be proficient in math, ou need to check our answers to problems using a different method and continuall ask ourself, Does this make sense? Work with a partner. Solve each equation b graphing. a. = 0 b. + = 0 c. + = 0 d. + = 0 e. + 5 = 0 f. + = 0 Communicate Your Answer. How can ou use a graph to solve a quadratic equation in one variable?. After ou find a solution graphicall, how can ou check our result algebraicall? Check our solutions for parts (a) (d) in Eploration algebraicall. 5. How can ou determine graphicall that a quadratic equation has no solution? Section 9. b Graphing 89

2 9. Lesson What You Will Learn Solve quadratic equations b graphing. Use graphs to find and approimate the zeros of functions. Core Vocabul Vocabular larr Solve real-life problems using graphs of quadratic functions. quadratic equation, p. 90 Previous -intercept root zero of a function b Graphing A quadratic equation is a nonlinear equation that can be written in the standard form a + b + c = 0, where a 0. In Chapter 7, ou solved quadratic equations b factoring. You can also solve quadratic equations b graphing. Core Concept b Graphing Step Write the equation in standard form, a + b + c = 0. Step Graph the related function = a + b + c. Step Find the -intercepts, if an. The solutions, or roots, of a + b + c = 0 are the -intercepts of the graph. Solving a Quadratic Equation: Two Real Solutions Solve + = b graphing. Step Write the equation in standard form. + = Write original equation. + = 0 Subtract from each side. Step Graph the related function = +. Step Find the -intercepts. The -intercepts are and. So, the solutions are = and =. = + Check + =? ( ) + ( ) = = Original equation Substitute. Monitoring Progress Simplif. + =? + () = = Help in English and Spanish at BigIdeasMath.com Solve the equation b graphing. Check our solutions.. = 0 90 Chapter = 0. + =

3 Solving a Quadratic Equation: One Real Solution Solve 8 = b graphing. Step Write the equation in standard form. ANOTHER WAY 8 = You can also solve the equation in Eample b factoring. Write original equation. 8 + = 0 Add to each side. Step Graph the related function = = 0 Step Find the -intercept. The onl -intercept is at the verte, (, 0). ( )( ) = 0 So, =. So, the solution is =. = 8 + Solving a Quadratic Equation: No Real Solutions Solve = + b graphing. Method Write the equation in standard form, + + = 0. Then graph the related function = + +, as shown at the left. There are no -intercepts. So, = + has no real solutions. Method Graph each side of the equation. = + + = Left side = + Right side = + The graphs do not intersect. So, = + has no real solutions. Monitoring Progress = Help in English and Spanish at BigIdeasMath.com Solve the equation b graphing.. + = 5. + = = 5 7. = = = Concept Summar Number of Solutions of a Quadratic Equation A quadratic equation has: two real solutions when the graph of its related function has two -intercepts. one real solution when the graph of its related function has one -intercept. no real solutions when the graph of its related function has no -intercepts. Section 9. b Graphing 9

4 Finding Zeros of Functions Recall that a zero of a function is an -intercept of the graph of the function. Finding the Zeros of a Function The graph of f () = ( )( ) is shown. Find the zeros of f. The -intercepts are,, and. So, the zeros of f are,, and. Check ) = 0 f () = ( )( ) = 0 f ( ) = ( )[( ) ( ) ] = 0 f () = ( f() = ( )( ) )( The zeros of a function are not necessaril integers. To approimate zeros, analze the signs of function values. When two function values have different signs, a zero lies between the -values that correspond to the function values. Approimating the Zeros of a Function The graph of f () = + + is shown. Approimate the zeros of f to the nearest tenth. There are two -intercepts: one between and, and another between and 0. Make tables using -values between and, and between and 0. Use an increment of 0.. Look for a change in the signs of the function values. f() = f () change in signs ANOTHER WAY You could approimate one zero using a table and then use the ais of smmetr to find the other zero f () The function values that are closest to 0 correspond to -values that best approimate the zeros of the function. change in signs In each table, the function value closest to 0 is 0.. So, the zeros of f are about.7 and 0.. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 0. Graph f () = +. Find the zeros of f.. Graph f () = + +. Approimate the zeros of f to the nearest tenth. 9 Chapter 9

5 Solving Real-Life Problems Real-Life Application A football plaer kicks a football feet above the ground with an initial vertical velocit of 75 feet per second. The function h = t + 75t + represents the height h (in feet) of the football after t seconds. (a) Find the height of the football each second after it is kicked. (b) Use the results of part (a) to estimate when the height of the football is 50 feet. (c) Using a graph, after how man seconds is the football 50 feet above the ground? Seconds, t Height, h a. Make a table of values starting with t = 0 seconds using an increment of. Continue the table until a function value is negative. The height of the football is feet after second, 88 feet after seconds, 8 feet after seconds, and feet after seconds. b. From part (a), ou can estimate that the height of the football is 50 feet between 0 and second and between and seconds. Based on the function values, it is reasonable to estimate that the height of the football is 50 feet slightl less than second and slightl less than seconds after it is kicked. c. To determine when the football is 50 feet above the ground, find the t-values for which h = 50. So, solve the equation t + 75t + = 50 b graphing. Step Write the equation in standard form. t + 75t + = 50 Write the equation. t + 75t 8 = 0 REMEMBER Subtract 50 from each side. Step Use a graphing calculator to graph the related function h = t + 75t Equations have solutions, or roots. Graphs have -intercepts. Functions have zeros. h = t + 75t 8 0 Step Use the zero feature to find the zeros of the function. 50 Zero X= Y=0 Zero X= Y=0 The football is 50 feet above the ground after about 0.8 second and about.9 seconds, which supports the estimates in part (b). Monitoring Progress Help in English and Spanish at BigIdeasMath.com. WHAT IF? After how man seconds is the football 5 feet above the ground? Section 9. b Graphing 9

6 Eercises 9. Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY What is a quadratic equation?. WHICH ONE DOESN T BELONG? Which equation does not belong with the other three? Eplain our reasoning. + 5 = 0 + = 0 = 7 + =. WRITING How can ou use a graph to find the number of solutions of a quadratic equation?. WRITING How are solutions, roots, -intercepts, and zeros related? Monitoring Progress and Modeling with Mathematics In Eercises 5 8, use the graph to solve the equation = = 0 0. =. =. 5 =. =. + = 8 5. ERROR ANALYSIS Describe and correct the error in 9. = 5 = + 8 = + + solving + = 8 b graphing = 0 8. = 0 = + = = = 0. = =. ERROR ANALYSIS Describe and correct the error in solving = 0 b graphing. 8 In Eercises, solve the equation b graphing. (See Eamples,, and.). 5 = 0. + = = 0. 7 = 0 7. = 9 8. = = Chapter 9 The solutions of the equation + = 8 are = and = 0. In Eercises 9, write the equation in standard form. 9. = The solution of the equation = 0 is = 9.

7 7. MODELING WITH MATHEMATICS The height.. (in ards) of a flop shot in golf can be modeled b = + 5, where is the horizontal distance (in ards). a. Interpret the -intercepts of the graph of the equation. f() = ( + )( ) f() = ( )( + ) b. How far awa does the golf ball land? 8. MODELING WITH MATHEMATICS The height h (in feet) of an underhand volleball serve can be modeled b h = t + 0t +, where t is the time (in seconds). In Eercises, approimate the zeros of f to the nearest tenth. (See Eample 5.). 0. =. 5 7 =. = 5. + = = = 8. f() = f() = +. f() = + +. = In Eercises 7, find the zero(s) of f. (See Eample.) 7. 5 b. No one receives the serve. After how man seconds does the volleball hit the ground? 9. = 0 a. Do both t-intercepts of the graph of the function have meaning in this situation? Eplain. In Eercises 9, solve the equation b using Method from Eample.. f() = + In Eercises 7 5, graph the function. Approimate the zeros of the function to the nearest tenth, if necessar. 7. f () = f () = + 9. = = f() = ( )( + ) 9. f() = ( + )( + + 8) 5. f () = MODELING WITH MATHEMATICS At a Civil War reenactment, a cannonball is fired into the air with an initial vertical velocit of 8 feet per second. The release point is feet above the ground. The function h = t + 8t + represents the height h (in feet) of the cannonball after t seconds. (See Eample.) a. Find the height of the cannonball each second after it is fired. f() = ( + )( + ) f () = + 5 b. Use the results of part (a) to estimate when the height of the cannonball is 50 feet. 8 c. Using a graph, after how man seconds is the cannonball 50 feet above the ground? f() = ( 5)( + ) Section 9. b Graphing 95

8 5. MODELING WITH MATHEMATICS You throw a softball straight up into the air with an initial vertical velocit of 0 feet per second. The release point is 5 feet above the ground. The function h = t + 0t + 5 represents the height h (in feet) of the softball after t seconds. 59. COMPARING METHODS Eample shows two methods for solving a quadratic equation. Which method do ou prefer? Eplain our reasoning. 0. THOUGHT PROVOKING How man different a. Find the height of the softball each second after it is released. parabolas have and as -intercepts? Sketch eamples of parabolas that have these two -intercepts. b. Use the results of part (a) to estimate when the height of the softball is 5 feet.. MODELING WITH MATHEMATICS To keep water off a road, the surface of the road is shaped like a parabola. A cross section of the road is shown in the diagram. The surface of the road can be modeled b = , where and are measured in feet. Find the width of the road to the nearest tenth of a foot. c. Using a graph, after how man seconds is the softball 5 feet above the ground? MATHEMATICAL CONNECTIONS In Eercises 55 and 5, use the given surface area S of the clinder to find the radius r to the nearest tenth. 55. S = 5 ft 5. S = 750 m r r m ft. MAKING AN ARGUMENT A stream of water from a fire hose can be modeled b = , where and are measured in feet. A firefighter is standing 57 feet from a building and is holding the hose feet above the ground. The bottom of a window of the building is feet above the ground. Your friend claims the stream of water will pass through the window. Is our friend correct? Eplain. 57. WRITING Eplain how to approimate zeros of a function when the zeros are not integers. 58. HOW DO YOU SEE IT? Consider the graph shown. REASONING In Eercises 5, determine whether the statement is alwas, sometimes, or never true. Justif our answer. = +. The graph of = a + c has two -intercepts when 8 = a is negative.. The graph of = a + c has no -intercepts when a and c have the same sign. a. How man solutions does the quadratic equation = + have? Eplain. 5. The graph of = a + b + c has more than two -intercepts when a 0. b. Without graphing, describe what ou know about the graph of = +. Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Determine whether the table represents an eponential growth function, an eponential deca function, or neither. Eplain. (Section.) Chapter