Solve Quadratic Equations by Graphing

Transcription

1 0.3 Solve Quadratic Equations b Graphing Before You solved quadratic equations b factoring. Now You will solve quadratic equations b graphing. Wh? So ou can solve a problem about sports, as in Eample 6. Ke Vocabular quadratic equation -intercept, p. 225 roots, p. 575 zero of a function, p. 337 A quadratic equation is an equation that can be written in the standard form a 2 b c 5 0 where a Þ 0. In Chapter 9, ou used factoring to solve a quadratic equation. You can also use graphing to solve a quadratic equation. Notice that the solutions of the equation a 2 b c 5 0 are the -intercepts of the graph of the related function 5 a 2 b c. Solve b Factoring Solve b Graphing ( 2 )( 2 5) or 5 5 To solve , graph From the graph ou can see that the -intercepts are and READING In this course, solutions refers to real-number solutions. To solve a quadratic equation b graphing, first write the equation in standard form, a 2 b c 5 0. Then graph the related function 5 a 2 b c. The -intercepts of the graph are the solutions, or roots, of a 2 b c 5 0. E XAMPLE Solve a quadratic equation having two solutions Solve b graphing. STEP Write the equation in standard form Write original equation Subtract 3 from each side. STEP 2 Graph the function The -intercepts are 2 and 3. c The solutions of the equation are 2 and 3. CHECK You can check 2 and 3 in the original equation Write original equation. (2) 2 2 2(2) 0 3 (3) 2 2 2(3) 0 3 Substitute for Simplif. Each solution checks. 0.3 Solve Quadratic Equations b Graphing 643

2 E XAMPLE 2 Solve a quadratic equation having one solution Solve b graphing. STEP Write the equation in standard form Write original equation Subtract from each side. STEP 2 Graph the function The -intercept is. c The solution of the equation is E XAMPLE 3 Solve a quadratic equation having no solution AVOID ERRORS Do not confuse -intercepts and -intercepts. Although the graph has a -intercept, it does not have an -intercepts. Solve b graphing. STEP Write the equation in standard form Write original equation Subtract 4 from each side. STEP 2 Graph the function The graph has no -intercepts. c The equation has no solution GUIDED PRACTICE for Eamples, 2, and 3 Solve the equation b graphing KEY CONCEPT For Your Notebook Number of s of a Quadratic Equation A quadratic equation has two solutions if the graph of its related function has two -intercepts. A quadratic equation has one solution if the graph of its related function has one -intercept. A quadratic equation has no real solution if the graph of its related function has no -intercepts. 644 Chapter 0 Quadratic Equations and Functions

3 FINDING ZEROS Because a zero of a function is an -intercept of the function s graph, ou can use the function s graph to find the zeros of a function. E XAMPLE 4 Find the zeros of a quadratic function Find the zeros of f() ANOTHER WAY You can find the zeros of a function b factoring: f() ( 7)( 2 ) 527 or 5 Graph the function f() The -intercepts are 27 and. c The zeros of the function are 27 and. CHECK Substitute 27 and in the original function. f(27) 5 (27) 2 6(27) f () f() 5 () 2 6() APPROXIMATING ZEROS The zeros of a function are not necessaril integers. To approimate zeros, look at the signs of the function values. If two function values have opposite signs, then a zero falls between the -values that correspond to the function values. E XAMPLE 5 Approimate the zeros of a quadratic function Approimate the zeros of f() to the nearest tenth. STEP Graph the function f() There are two -intercepts: one between 24 and 23 and another between 2 and 0. STEP 2 Make a table of values for -values between 24 and 23 and between 2 and 0 using an increment of 0.. Look for a change in the signs of the function values. f () INTERPRET FUNCTION VALUES The function value that is closest to 0 indicates the -value that best approimates a zero of the function f() f() c In each table, the function value closest to 0 is 20.. So, the zeros of f() are about 23.7 and about GUIDED PRACTICE for Eamples 4 and 5 4. Find the zeros of f() Approimate the zeros of f() to the nearest tenth. 0.3 Solve Quadratic Equations b Graphing 645

4 E XAMPLE 6 Solve a multi-step problem SPORTS An athlete throws a shot put with an initial vertical velocit of 40 feet per second as shown. a. Write an equation that models the height h (in feet) of the shot put as a function of the time t (in seconds) after it is thrown. b. Use the equation to find the time that the shot put is in the air. 6.5 ft a. Use the initial vertical velocit and the release height to write a vertical motion model. h 526t 2 vt s Vertical motion model h 526t 2 40t 6.5 Substitute 40 for v and 6.5 for s. USE A GRAPHING CALCULATOR When entering h 526t 2 40t 6.5 in a graphing calculator, use instead of h and instead of t. b. The shot put lands when h 5 0. To find the time t when h 5 0, solve 0 526t 2 40t 6.5 for t. To solve the equation, graph the related function h 526t 2 40t 6.5 on a graphing calculator. Use the trace feature to find the t-intercepts. c There is onl one positive t-intercept. The shot put is in the air for about 2.6 seconds. Trace X= Y= GUIDED PRACTICE for Eample 6 6. WHAT IF? In Eample 6, suppose the initial vertical velocit is 30 feet per second. Find the time that the shot put is in the air. CONCEPT SUMMARY For Your Notebook Relating s of Equations, -Intercepts of Graphs, and Zeros of Functions s of an Equation The solutions of the equation are 2 and 6. -Intercepts of a Graph The -intercepts of the graph of occur where 5 0, so the -intercepts are 2 and 6, as shown. Zeros of a Function The zeros of the function f() are the values of for which f() 5 0, so the zeros are 2 and Chapter 0 Quadratic Equations and Functions

5 0.3 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5 and 5 5 STANDARDIZED TEST PRACTICE Es. 2, 46, 53, and 54. VOCABULARY Write in standard form. 2. WRITING Is a quadratic equation? Eplain. SOLVING EQUATIONS Solve the equation b graphing. EXAMPLES, 2, and 3 on pp for Es } } } ERROR ANALYSIS The graph of the function related to the equation is shown. Describe and correct the error in solving the equation. The onl solution of the equation is 4. 3 EXAMPLE 4 on p. 645 for Es FINDING ZEROS Find the zeros of the function. 22. f() f() f() f() f() f() f() f() f() SOLVING EQUATIONS Solve the equation b graphing } EXAMPLE 5 on p. 645 for Es APPROXIMATING ZEROS Approimate the zeros of the function to the nearest tenth. 37. f() f() f() f() f() f() f() f() f() MULTIPLE CHOICE Which function has a zero between 23 and 22? A f() B f() C f() D f() Solve Quadratic Equations b Graphing 647

6 CHALLENGE Use the given surface area S of the clinder to find the radius r to the nearest tenth. (Use 3.4 for p.) 47. S 5 25 ft S 5 76 m S cm 2 r 6 ft r 3 m r 0 cm PROBLEM SOLVING GRAPHING CALCULATOR You ma wish to use a graphing calculator to complete the following Problem Solving eercises. EXAMPLE 6 on p. 646 for Es SOCCER The height (in feet) of a soccer ball after it is kicked can be modeled b the graph of the equation where is the horizontal distance (in feet) that the ball travels. The ball is not touched, and it lands on the ground. Find the distance that the ball was kicked. 5. SURVEYING To keep water off a road, the road s surface is shaped like a parabola as in the cross section below. The surface of the road can be modeled b the graph of where and are measured in feet. Find the width of the road to the nearest tenth of a foot DIVING During a cliff diving competition, a diver begins a dive with his center of gravit 70 feet above the water. The initial vertical velocit of his dive is 8 feet per second. a. Write an equation that models the height h (in feet) of the diver s center of gravit as a function of time t (in seconds). b. How long after the diver begins his dive does his center of gravit reach the water? 53. SHORT RESPONSE An arc of water spraed from the nozzle of a fountain can be modeled b the graph of where is the horizontal distance (in feet) from the nozzle and is the vertical distance (in feet). The diameter of the circle formed b the arcs on the surface of the water is called the displa diameter. Find the displa diameter of the fountain. Eplain our reasoning. Displa diameter WORKED-OUT SOLUTIONS on p. WS 5 STANDARDIZED TEST PRACTICE

7 54. EXTENDED RESPONSE Two softball plaers are practicing catching fl balls. One plaer throws a ball to the other. She throws the ball upward from a height of 5.5 feet with an initial vertical velocit of 40 feet per second for her teammate to catch. a. Write an equation that models the height h (in feet) of the ball as a function of time t (in seconds) after it is thrown. b. If her teammate misses the ball and it lands on the ground, how long was the ball in the air? c. If her teammate catches the ball at a height of 5.5 feet, how long was the ball in the air? Eplain our reasoning. 55. CHALLENGE A stream of water from a fire hose can be modeled b the graph of where and are measured in feet. A firefighter is holding the hose 3 feet above the ground, 37 feet from a building. Will the stream of water pass through a window if the top of the window is 26 feet above the ground? Eplain. MIXED REVIEW PREVIEW Prepare for Lesson 0.4 in Es Evaluate the epression. (p. 0) 56. 2Ï } Ï } Ï } 625 Simplif the epression. (p. 495) (28) } 60. (28) p 9 6 } } Write the number in standard form. (p. 52) QUIZ for Lessons Graph the function. Compare the graph with the graph of 5 2. (p. 628). 52} Graph the function. Label the verte and ais of smmetr (p. 628) (p. 628) (p. 635) (p. 635) 8. 52} (p. 635) (p. 635) Solve the equation b graphing. (p. 643) Find the zeros of the function. (p. 643) 6. f() f() f() EXTRA PRACTICE for Lesson 0.3, p. 947 ONLINE QUIZ at classzone.com 649