Solving Quadratic Equations by Graphing 9.1. ACTIVITY: Solving a Quadratic Equation by Graphing. How can you use a graph to solve a quadratic

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1 9. Solving Quadratic Equations b Graphing equation in one variable? How can ou use a graph to solve a quadratic Earlier in the book, ou learned that the -intercept of the graph of = a + b variables is the same as the solution of a + b = 0. variable The -intercept of the graph of is. (, 0) 7 The solution of the equation 0 is. ACTIVITY: Solving a Quadratic Equation b Graphing Work with a partner. CMMN CRE Solving Quadratic Equations In this lesson, ou will solve quadratic equations b graphing. Learning Standards A.REI.b A.REI. a. Sketch the graph of =. 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 that the -values found in part (c) are solutions of = Chapter 9 Solving Quadratic Equations

2 Math Practice Use Clear Definitions How is the solution of the equation represented b the graph of the equation? ACTIVITY: Solving Quadratic Equations b Graphing Work with a partner. Solve each equation b graphing. a. = 0 b. + = 0 c. + = 0 d. + = IN YUR WN WRDS 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? Use our solutions in Activit as eamples. Use what ou learned about solving quadratic equations to complete Eercises 7 on page 9. Section 9. Solving Quadratic Equations b Graphing

3 9. Lesson Lesson Tutorials Ke Vocabular quadratic equation, p. 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 in standard form b finding the -intercept(s) of the graph of the related function = a + b + c. EXAMPLE Solving a Quadratic Equation: Two Real Solutions Remember The solutions of a quadratic equation are also called roots. Solve + = 0 b graphing. Step : Graph the related function = +. Step : Find the -intercepts. The are and. So, the solutions are = and =. Check Check each solution in the original equation. + = 0 riginal equation + = 0 ( ) + ( ) =? 0 Substitute. + () =? 0 0 = 0 Simplif. 0 = 0 EXAMPLE Solving a Quadratic Equation: ne Real Solution Stud Tip You can also solve the equation in Eample b factoring. 8 + = 0 ( )( ) = 0 So, =. Solve 8 = b graphing. Step : Rewrite the equation in standard form. 8 = 8 + = 0 Step : Graph the related function = 8 +. Step : Find the -intercept. The onl -intercept is at the verte (, 0). So, the solution is =. Write the equation. Add to each side. 8 Solve the equation b graphing. Check our solution(s). Eercises 8 0. = = 0. + =. + =. + = = Chapter 9 Solving Quadratic Equations

4 EXAMPLE Solving a Quadratic Equation: No Real Solutions Solve = + b graphing. Method : Rewrite the equation in standard form and graph the related function = + +. 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. Eercises Solve the equation b graphing. Check our solution(s). 7. = = 9. + = Quadratic equations ma have two real solutions, one real solution, or no real solutions. two real solutions one real solution no real solutions two -intercepts one -intercept no -intercepts Section 9. Solving Quadratic Equations b Graphing 7

5 EXAMPLE Real-Life Application A football plaer kicks a football feet above the ground with an upward velocit of 7 feet per second. The function h = t + 7t + gives the height h (in feet) of the football after t seconds. After how man seconds is the football 0 feet above the ground? To determine when the football is 0 feet above the ground, find the t-values for which h = 0. So, solve the equation t + 7t + = 0. Step : Rewrite the equation in standard form. t + 7t + = 0 t + 7t 8 = 0 Step : Use a graphing calculator to graph the related function h = t + 7t 8. Write the equation. Subtract 0 from each side. 0 h t 7t 8 Step : Use the zero feature to find the zeros of the function Remember A zero of a function = f () is an -value for which the value of the function is zero. 0 The football is 0 feet above the ground after about 0.8 second and about.9 seconds. 0 Eercise 8 0. WHAT IF? After how man seconds is the football feet above the ground? The solutions, or roots, of + = 0 are = and =. The -intercepts of the graph of = + are and. The zeros of the function f () = + are and. 8 Chapter 9 Solving Quadratic Equations

6 9. Eercises Help with Homework. VCABULARY What is a quadratic equation?. WHICH NE DESN T BELNG? Which equation does not belong with the other three? Eplain our reasoning. + = 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? 9+(-)= +(-)= +(-9)= 9+(-)= Determine the solution(s) of the equation. Check our solution(s) = 0. = = Solve the equation b graphing. Check our solution(s). 8. = = = 0. + = 0. + = = 0. + = = = 0 7. FLP SHT The height (in ards) of a flop shot in golf can be modeled b = +, where is the horizontal distance (in ards). a. Interpret the -intercepts of the graph of the equation. b. How far awa does the golf ball land? 8. VLLEYBALL The height h (in feet) of an underhand volleball serve can be modeled b h = t + 0t +, where t is the time in seconds. After how man seconds is the ball feet above the ground? Section 9. Solving Quadratic Equations b Graphing 9

7 Rewrite the equation in standard form. Then solve the equation b graphing. Check our solution(s) with a graphing calculator. 9. = 8 0. =. =. =. =. 8 = Solve the equation b using Method from Eample. Check our solution(s).. = 0. = 7. 7 = 8. = 0 9. = 0. 8 = 9. REASNING Eample shows two methods for solving a quadratic equation. Which method do ou prefer? Eplain our reasoning.. ERRR ANALYSIS Describe and correct the error in solving the equation The onl solution of the equation = 0 is = BASEBALL A baseball plaer throws a baseball with an upward velocit of feet per second. The release point is feet above the ground. The function h = t + t + gives the height h (in feet) of the baseball after t seconds. a. How long is the ball in the air if no one catches it? b. How long does the ball remain above feet?. SFTBALL You throw a softball straight up into the air with an upward velocit of 0 feet per second. The release point is feet above the ground. The function h = t + 0t + gives the height h (in feet) of the softball after t seconds. a. How long is the ball in the air if ou miss it? b. How long is the ball in the air if ou catch it at a height of feet? 0 Chapter 9 Solving Quadratic Equations

8 Use a graphing calculator to approimate the zeros of the function to the nearest tenth.. f () = + +. f () = 7. f () = + 8. f () = + 9. f () = + 0. f () = + 9. MDELING A dirt bike launches off a ramp that is 8 feet tall. The upward velocit of the dirt bike is 0 feet per second. a. Write a function that models the height h (in feet) of the dirt bike after t seconds. b. After how man seconds does the dirt bike land?. WRLD S STRNGEST MAN ne of the events in the World s Strongest Man competition is the keg toss. In this event, competitors tr to throw kegs of various weights over a wall that is feet inches high. a. A competitor releases a keg feet above the ground with an upward velocit of 7 feet et per second. Is this throw high enough to clear the wall? Eplain our reasoning. b. Do the heights of the competitors factor into their success at this event? Eplain our reasoning. Determine whether the statement is sometimes, alwas, or never true. Justif our answer.. The graph of = a + c has two -intercepts when a =.. The graph of = a + c has no -intercepts when a and c have the same sign.. The graph of = a + b + c has more than two zeros when a 0. Simplif the epression. (Section.) MULTIPLE CHICE Which epression is equivalent to ( m )? (Section.) A m 7 B m 0 C 9m 7 D 9m 0 Section 9. Solving Quadratic Equations b Graphing