20.2 Connecting Intercepts and Linear Factors

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1 Name Class Date 20.2 Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? Resource Locker Eplore Connecting Factors and Intercepts Use graphs and linear factors to find the intercepts of a parabola. A Graph = + and = - 2 using a graphing calculator. Then sketch the graphs on the grid. 8 B Identif the -intercept of each line. The -intercepts are and C The quadratic function = ( + ) ( - 2) is the product of the two linear factors that have been graphed. Use a graphing calculator to graph the function = ( + ) ( - 2). Then sketch a graph of the quadratic function on the same grid with the linear factors that have been graphed. - D Identif the -intercepts of the parabola. The -intercepts are and. E What do ou notice about the intercepts of the parabola? Reflect 1. Use a graph to determine whether is the product of the linear factors 2-3 and Discussion Make a conjecture about the linear factors and -intercepts of a quadratic function Module Lesson 2

2 Eplain 1 Rewriting from Factored Form to Standard Form A quadratic function is in factored form when it is written as = k ( - a) ( - b) where k 0. Eample 1 A Write each function in standard form. = 2 ( + 1) ( - ) Multipl the two linear factors. = 2 ( ) = 2 ( ) Multipl the resulting trinomial b 2. = The standard form of = 2 ( + 1) ( - ) is = B = 3 ( - 5) ( - 2) Multipl the two linear factors. = 3 ( ) ( ) = 3 ( ) Multipl the resulting trinomial b 3. = The standard form of = 3 ( - 5) ( - 2) is. Reflect For the function = k( - a)( - b) when k > 0, a 0, and b 0, consider these questions. 3. For the standard form of the function, will the coefficient of the -term be positive or negative if a and b are both positive? If a and b are both negative?. For the standard form of the function, will the constant term be positive or negative if a and b have the same sign? If a and b have different signs? Your Turn Write each function in standard form. 5. = ( - 7) ( - 1) 6. = ( - 1) ( + 3) Module Lesson 2

3 Eplain 2 Connecting Factors and Zeros In the Eplore ou learned that the factors in factored form indicate the -intercepts of a function. In a previous lesson ou learned that the -intercepts of a graph are the zeros of the function. Eample 2 Write each function in standard form. Determine -intercepts and zeros of each function. A = 2 ( - 1) ( - 3) Write the function in standard form. The factors indicate the intercepts. * Factor ( 1) indicates an intercept of 1. * Factor ( 3) indicates an intercept of 3. The -intercepts of a graph are the zeros of the function. = 2 ( ) = 2 ( ) = * An intercept of 1 indicates that the function has a zero of 1. * An intercept of 3 indicates that the function has a zero of 3. B = 2 ( + ) ( + 2) Write the function in standard form. The factors indicate the intercepts. * Factor ( + ) indicates an intercept of. * Factor indicates an intercept of 2. The intercepts of a graph are the zeros of the function. = 2 ( ) ( ) = 2 = * An intercept of indicates that the function has a zero of. * An intercept of indicates that the function has a zero of 2. Reflect 7. Discussion What are the zeros of a function? 8. How man -intercepts can quadratic functions have? Module Lesson 2

4 Your Turn Write each function in standard form. Determine intercepts and zeros of each function. 9. = -2 ( + 5) ( + 1) 10. = 5 ( - 3) ( - 1) Eplain 3 Writing Quadratic Functions Given -Intercepts Given two quadratic functions ƒ () = ( - a) ( - b) and g () = k ( - a) ( - b), where k is an non-zero real constant, eamine the intercepts for each quadratic function. f () = ( - a) ( - b) 0 = ( - a) ( - b) - a = 0 or - b =0 = a = b g () = k ( - a) ( - b) 0 = k ( - a) ( - b) 0 = ( - a) ( - b) - a = 0 or - b = 0 = a = b Notice that ƒ () = ( - a) ( - b) and g () = k ( - a) ( - b) have the same -intercepts. You can use the factored form to construct a quadratic function given the intercepts and the value of k. Eample 3 For the two given intercepts, use the factored form to generate a quadratic function for each given constant k. Write the function in standard form. A -intercepts: 2 and 5; k = 1, k = -2, k =3 Write the quadratic function with k = 1 using ƒ () = k ( - a) ( - b). ƒ () = 1 ( - 2) ( - 5) ƒ () = ( - 2) ( - 5) 2 ƒ () = Write the quadratic function with k = -2. ƒ () = -2( - 2) ( - 5) ƒ () = -2( ) ƒ () = Write the quadratic function with k = 3. ƒ () = 3 ( - 2) ( - 5) ƒ () = 3 ( ) ƒ () = Module Lesson 2

5 B -intercepts: -3 and ; k = 1, k = -3, k = 2 Write the quadratic function with k = 1. ƒ () = ƒ () = Write the quadratic function with k = -3. ƒ () = ƒ () = Write the quadratic function with k = 2. ƒ () = ƒ () = Reflect 11. How are the functions with same intercepts but different constant factors the same? How are the different? Your Turn For the given two intercepts and three values of k generate three quadratic functions. Write the functions in factored form and standard form intercepts: 1 and 8; k = 1, k = -, k = intercepts: -7 and 3; k = 1, k = -5, k = 7 Module Lesson 2

6 Elaborate 1. If the intercepts of a quadratic function are 3 and 8, what can be said about the intercepts of its linear factors? 15. If a quadratic function has onl one zero, it has to occur at the verte of the parabola. Using the graph of a quadratic function, eplain wh. 16. How are intercepts and zeros related? 17. What would the factored form look like if there were onl one intercept? 18. Essential Question Check-In How can ou find intercepts of a quadratic function if its linear factors are known? Module Lesson 2

7 Evaluate: Homework and Practice Graph each quadratic function and each of its linear factors. Then identif the -intercepts and the ais of smmetr of each parabola. Online Homework Hints and Help Etra Practice 1. = ( - 2) ( - 6) 2. = ( + 3) ( - 1) = ( - 5) ( + 2). = ( - 5) ( - 5) Module Lesson 2

8 Write each function in standard form. 5. = 5 ( - 2) ( + 1) 6. = 2 ( + 6) ( + 3) 7. = -2 ( + ) ( - 5) 8. = - ( + 2) ( + 3) 9. Which of the following is the correct standard form of = 3 ( - 8) ( - 5)? a. = b. = c. = d. = e. = The area of a Japanese rock garden is = 7 ( - 3) ( + 1). Write = 7 ( - 3) ( + 1) in standard form. Write each function in standard form. Determine intercepts and zeros of each function. 11. = - (2 - ) ( - 2) 12. = 2 ( + ) ( - 2) 13. = -3 ( + 1) ( - 3) 1. = 2 ( + 2) ( - 1) Image Credits: David Maska/Shutterstock Module Lesson 2

9 15. A soccer ball is kicked from ground level. The function = -16 ( - 2) gives the height (in feet) of the ball, where is time (in seconds). After how man seconds will the ball hit the ground? Use a graphing calculator to verif our answer. 16. A tennis ball is tossed upward from a balcon. The height of the ball in feet can be modeled b the function = - (2 + 1) (2-3) where is the time in seconds after the ball is released. Find the maimum height of the ball and the time it takes the ball to reach this height. Determine how long it takes the ball to hit the ground. For the two given intercepts, use the factored form to generate a quadratic function for each given constant k. Write the function in standard form intercepts: -5 and 3; k = 1, k = -2, k = intercepts: and 7; k = 1, k = -3, k = 5 H.O.T. Focus on Higher Order Thinking 19. Eplain the Error For the given two intercepts, 3 and 9, k =, Kell wrote a quadratic function in factored form, ƒ () = ( + 3) ( + 9), and in standard form, f () = What error did she make? 20. Critical Thinking How is the graph of ƒ () = 7 ( + 3) ( - 2) similar to and different from the graph of ƒ () = ? 21. Make a Prediction How could ou find an equation of a quadratic function with zeros at 3 and at 1? Module Lesson 2

10 Lesson Performance Task The cross-sectional shape of the archwa of a bridge (measured in feet) is modeled b the function ƒ () = where ƒ () is the height of the arch and is the horizontal distance from one side of the base. How wide is the arch at its base? Will a bo truck that is 8 feet wide and 13.5 feet tall fit under the arch? If not, what is the maimum height a truck that is 8 feet wide and is passing under the bridge can be? Module Lesson 2