Name Date Chapter 8 Maintaining Mathematical Proficienc Graph the linear equation. 1. = 5. = + 3 3. 1 = + 3. = + Evaluate the epression when =. 5. + 8. + 3 7. 3 8. 5 + 8 9. 8 10. 5 + 3 11. + + 1. 3 + + Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8.1 Graphing f( ) = a For use with Eploration 8.1 Essential Question What are some of the characteristics of the graph of a quadratic function of the form f( ) = a? 1 EXPLORATION: Graphing Quadratic Functions Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Graph each quadratic function. Compare each graph to the graph of f () =. a. g() 3 = b. g() = 5 10 8 f() = f() = 8 1 1 c. g( ) 0. = d. g( ) = 1 10 f() = 10 8 f() = Copright Big Ideas Learning, LLC Algebra 1 5
Name Date 8.1 Graphing f( )= a (continued) Communicate Your Answer. What are some of the characteristics of the graph of a quadratic function of the form f () = a? 3. How does the value of a affect the graph of f () = a? Consider 0 < a < 1, a > 1, 1 < a < 0, and a < 1. Use a graphing calculator to verif our answers.. The figure shows the graph of a quadratic function of the form = a. Which of the intervals in Question 3 describes the value of a? Eplain our reasoning. 7 1 Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8.1 Notetaking with Vocabular For use after Lesson 8.1 In our own words, write the meaning of each vocabular term. quadratic function parabola verte ais of smmetr Core Concepts Characteristics of Quadratic Functions The parent quadratic function is f ( ) =. The graphs of all other quadratic functions are transformations of the graph of the parent quadratic function. The lowest point on a parabola that opens up or the highest point on a parabola that opens down is the verte. The verte of the graph of f ( ) = is (0, 0). decreasing ais of smmetr increasing verte The vertical line that divides the parabola into two smmetric parts is the ais of smmetr. The ais of smmetr passes through the verte. For the graph of f ( ) =, the ais of smmetr is the -ais, or = 0. Notes: Copright Big Ideas Learning, LLC Algebra 1 7
Name Date 8.1 Notetaking with Vocabular (continued) Graphing f( ) = a When a > 0 When 0 < a < 1, the graph of of f ( ) =. f ( ) = a is a vertical shrink of the graph a = 1 a > 1 When a > 1, the graph of f ( ) =. f ( ) = a is a vertical stretch of the graph of 0 < a < 1 Graphing f( ) = a When a < 0 When 1 < a < 0, the graph of f ( ) reflection in the -ais of the graph of When 1, f = a is a vertical shrink with a f ( ) =. = a is a vertical stretch with a a < the graph of ( ) reflection in the -ais of the graph of f ( ) =. Notes: a = 1 1 < a < 0 a < 1 Etra Practice In Eercises 1 and, identif characteristics of the quadratic function and its graph. 1.. 8 8 Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8.1 Notetaking with Vocabular (continued) In Eercises 3 8, graph the function. Compare the graph to the graph of f( ) =. 3. g( ) 5 =. m ( ) = 5. k ( ) =. l ( ) = 7 7. n ( ) = 1 8. 5 p( ) = 0. In Eercises 9 and 10, determine whether the statement is alwas, sometimes, or never true. Eplain our reasoning. 9. The graph of g( ) = a is wider than the graph of f ( ) = when a > 0. 10. The graph of g( ) = a is narrower than the graph of f ( ) = when a < 1. Copright Big Ideas Learning, LLC Algebra 1 9
Name Date 8. Graphing f( )= a + c For use with Eploration 8. Essential Question How does the value of c affect the graph of f( ) = a + c? 1 EXPLORATION: Graphing = a + c Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Sketch the graphs of the functions in the same coordinate plane. What do ou notice? a. f ( ) = and g ( ) = + 10 8 b. f ( ) = and g ( ) = 10 8 50 Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8. Graphing f( )= a + c (continued) EXPLORATION: Finding -Intercepts of Graphs Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Graph each function. Find the -intercepts of the graph. Eplain how ou found the -intercepts. a. = 7 b. = + 1 8 8 Communicate Your Answer 3. How does the value of c affect the graph of f ( ) a c = +?. Use a graphing calculator to verif our answers to Question 3. 5. The figure shows the graph of a quadratic function of the form = a + c. Describe possible values of a and c. Eplain our reasoning. 7 1 Copright Big Ideas Learning, LLC Algebra 1 51
Name Date 8. Notetaking with Vocabular For use after Lesson 8. In our own words, write the meaning of each vocabular term. zero of a function Core Concepts Graphing f( )= a + c When c > 0, the graph of f ( ) = a + cis a vertical translation c units up of the graph of f ( ) = a. c > 0 c = 0 When c < 0, the graph of f ( ) = a + cis a vertical translation c units down of the graph of f ( ) = a. The verte of the graph of f ( ) = a + c is ( c) smmetr is = 0. Notes: 0,, and the ais of c < 0 5 Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8. Notetaking with Vocabular (continued) Etra Practice In Eercises 1, graph the function. Compare the graph to the graph of f( ) =. 1. g ( ) 5 = +. m ( ) = 3 3. n =. q ( ) = 1 ( ) 3 Copright Big Ideas Learning, LLC Algebra 1 53
Name Date 8. Notetaking with Vocabular (continued) In Eercises 5 8, find the zeros of the function. 5. = + 1. = + 1 7. n ( ) = + 8. p ( ) = 9 + 1 In Eercises 9 and 10, sketch a parabola with the given characteristics. 9. The parabola opens down, and the verte 10. The lowest point on the parabola is (0, 5). is (0, ). 11. The function f () t = 1t + s0 represents the approimate height (in feet) of a falling object t seconds after it is dropped from an initial height s 0 (in feet). A tennis ball falls from a height of 00 feet. a. After how man seconds does the tennis ball hit the ground? b. Suppose the initial height is decreased b 38 feet. After how man seconds does the ball hit the ground? 5 Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8.3 Graphing f( )= a + b + c For use with Eploration 8.3 Essential Question How can ou find the verte of the graph of f( ) = a + b+ c? 1 EXPLORATION: Comparing -Intercepts with the Verte Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. a. Sketch the graphs of = 8 and = 8 +. 8 8 8 8 8 8 8 8 b. What do ou notice about the -coordinate of the verte of each graph? c. Use the graph of = 8 to find its -intercepts. Verif our answer b solving 0 = 8. d. Compare the value of the -coordinate of the verte with the values of the -intercepts. Copright Big Ideas Learning, LLC Algebra 1 55
Name Date 8.3 Graphing f( )= a + b+ c (continued) EXPLORATION: Finding -Intercepts Work with a partner. a. Solve 0 = a + b for b factoring. b. What are the -intercepts of the graph of = a + b? c. Complete the table to verif our answer. = a + b 0 b a. 3 EXPLORATION: Deductive Reasoning Work with a partner. Complete the following logical argument. b The -intercepts of the graph of = a + b are 0 and. a The verte of the graph of = a + b occurs when =. The vertices of the graphs of = a + b and = a + b + c have the same -coordinate. The verte of the graph of = a + b + c occurs when =. Communicate Your Answer. How can ou find the verte of the graph of f ( ) = a + b + c? 5. Without graphing, find the verte of the graph of f ( ) = + 3. Check our result b graphing. 5 Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8.3 Notetaking with Vocabular For use after Lesson 8.3 In our own words, write the meaning of each vocabular term. maimum value minimum value Core Concepts Graphing f( )= a + b + c The graph opens up when a > 0, down when a < 0. and the graph opens = b a f() = a + b + c, where a > 0 The -intercept is c. b The -coordinate of the verte is. a b The ais of smmetr is =. a (0, c) verte Notes: Copright Big Ideas Learning, LLC Algebra 1 57
Name Date 8.3 Notetaking with Vocabular (continued) Maimum and Minimum Values The -coordinate of the verte of the graph of f ( ) = a + b + c is the maimum value of the function when a < 0 or the minimum value of the function when a > 0. f ( ) = a + b + c, a < 0 f ( ) = a + b + c, a > 0 maimum minimum Notes: Etra Practice In Eercises 1, find (a) the ais of smmetr and (b) the verte of the graph of the function. 1. f( ) = 10 +. = + 1 3. = 8 + 5. f ( ) = 3 + + 1 58 Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8.3 Notetaking with Vocabular (continued) In Eercises 5 7, graph the function. Describe the domain and range. 5. f ( ) = 3 + +. = 8 1 7. = 1 + 5 5 In Eercises 8 13, tell whether the function has a minimum value or a maimum value. Then find the value. 8. = 1 5 + 9. = 8 + 1 10. = 7 11. = 7 + 7 + 5 1. = 9 + + 13. = 1 + 1. The function h = 1t + 50t represents the height h (in feet) of a rocket t seconds after it is launched. The rocket eplodes at its highest point. a. When does the rocket eplode? b. At what height does the rocket eplode? Copright Big Ideas Learning, LLC Algebra 1 59
Name Date f = a h + k 8. Graphing ( ) ( ) For use with Eploration 8. Essential Question How can ou describe the graph of f( ) = a( h )? 1 EXPLORATION: Graphing = a( h ) When h > 0 Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Sketch the graphs of the functions in the same coordinate plane. How does the value of h affect the graph of = a( h)? a. f () = and g () = ( ) 10 8 b. f () = and 10 g ( ) = ( ) 8 0 Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8. Graphing f( ) = a( h) + k (continued) EXPLORATION: Graphing = a( h ) When h < 0 Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Sketch the graphs of the functions in the same coordinate plane. How does the value of h affect the graph of = a ( h)? a. f ( ) = and g ( ) = ( + ) b. f ( ) = and g ( ) = ( + ) 8 8 Communicate Your Answer 3. How can ou describe the graph of f () = a( h)?. Without graphing, describe the graph of each function. Use a graphing calculator to check our answer. a. = ( 3) b. = ( + 3) c. = ( 3) Copright Big Ideas Learning, LLC Algebra 1 1
Name Date 8. Notetaking with Vocabular For use after Lesson 8. In our own words, write the meaning of each vocabular term. even function odd function verte form (of a quadratic function) Core Concepts Even and Odd Functions A function = f( ) is even when f ( ) f ( ) = for each in the domain of f. The graph of an even function is smmetric about the -ais. A function = f( ) is odd when f ( ) f ( ) = for each in the domain of f. The graph of an odd function is smmetric about the origin. A graph is smmetric about the origin when it looks the same after reflections in the -ais and then in the -ais. Notes: Graphing f( ) = a( h) When 0, h > the graph of f ( ) a( h) translation h units right of the graph = is a horizontal f ( ) = a. h = 0 When h < 0, the graph of translation h units left of the graph of The verte of the graph of f ( ) a( h) ais of smmetr is = h. Notes: f ( ) = a( h) is a horizontal f ( ) = a. = is ( h ),0, and the h < 0 h > 0 Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8. Notetaking with Vocabular (continued) Graphing f( ) = a( h) + k The verte form of a quadratic function is where a 0. The graph of ( ) = ( ) +, f a h k f ( ) = a( h) + k is a translation h units horizontall and k units verticall of the graph of f ( ) = a. The verte of the graph of f ( ) a( h) k the ais of smmetr is = h. = + is ( ) hk,, and f() = a (0, 0) h f() = a( h) + k (h, k) k Notes: Etra Practice In Eercises 1, determine whether the function is even, odd, or neither. 1. f ( ) = 5. f ( ) = 1 3. h ( ) =. ( ) = 3 + + 1 f In Eercises 5 8, find the verte and the ais of smmetr of the graph of the function. 5. f( ) = 5( ). f( ) = ( + 8) Copright Big Ideas Learning, LLC Algebra 1 3
Name Date 8. Notetaking with Vocabular (continued) 1 7. p ( ) = ( 1) + 8. g ( ) = ( + 1) 5 In Eercises 9 and 10, graph the function. Compare the graph to the graph of f( ) =. 9. m ( ) = 3( + ) 1 10. g ( ) = ( ) + In Eercises 11 and 1, graph g. 1 11. f( ) = 3( + 1) 1; g( ) = f( + ) 1. f ( ) = ( 3) 5; g( ) = f( ) Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8.5 Using Intercept Form For use with Eploration 8.5 Essential Question What are some of the characteristics of the graph of f( ) = a( p)( q )? 1 EXPLORATION: Using Zeros to Write Functions Work with a partner. Each graph represents a function of the form f () = ( p)( q) or f () = ( p)( q). Write the function represented b each graph. Eplain our reasoning. a. b. c. d. e. f. Copright Big Ideas Learning, LLC Algebra 1 5
Name Date 8.5 Using Intercept Form (continued) 1 EXPLORATION: Using Zeros to Write Functions (continued) g. h. Communicate Your Answer. What are some of the characteristics of the graph of f ( ) = a( p)( q)? 3. Consider the graph of f ( ) = a( p)( q). a. Does changing the sign of a change the -intercepts? Does changing the sign of a change the -intercept? Eplain our reasoning. b. Does changing the value of p change the -intercepts? Does changing the value of p change the -intercept? Eplain our reasoning. Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8.5 Notetaking with Vocabular For use after Lesson 8.5 In our own words, write the meaning of each vocabular term. intercept form Core Concepts Graphing f( ) = a( p)( q) The -intercepts are p and q. The ais of smmetr is halfwa between ( p,0) and ( ) p + q the ais of smmetr is =. q,0. So, = p + q The graph opens up when a > 0, and the graph opens down when a < 0. (p, 0) (q, 0) Notes: Factors and Zeros For an factor the polnomial. n of a polnomial, n is a zero of the function defined b Notes: Copright Big Ideas Learning, LLC Algebra 1 7
Name Date 8.5 Notetaking with Vocabular (continued) Etra Practice In Eercises 1 and, find the -intercepts and ais of smmetr of the graph of the function. 1. = ( + )( ). = 3( )( 3) In Eercises 3, graph the quadratic function. Label the verte, ais of smmetr, and -intercepts. Describe the domain and range of the function. 3. m ( ) = ( + 5)( + 1). = ( 3)( 1) 5. =. f ( ) = + 15 8 Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8.5 Notetaking with Vocabular (continued) In Eercises 7 and 8, find the zero(s) of the function. 7. = 8. = + 9 + 0 In Eercises 9 1, use zeros to graph the function. 9. f( ) = 3 10 10. f( ) = ( + 3)( 1) 11. 3 f ( ) 9 = 1. 3 f ( ) = 1 + 10 Copright Big Ideas Learning, LLC Algebra 1 9
Name Date 8. Comparing Linear, Eponential, and Quadratic Functions For use with Eploration 8. Essential Question How can ou compare the growth rates of linear, eponential, and quadratic functions? 1 EXPLORATION: Comparing Speeds Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Three cars start traveling at the same time. The distance traveled in t minutes is miles. Complete each table and sketch all three graphs in the same coordinate plane. Compare the speeds of the three cars. Which car has a constant speed? Which car is accelerating the most? Eplain our reasoning. t = t t t = 1 t = t 0 0 0 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.8 0.8 0.8 1.0 1.0 1.0 0.8 0. 0. 0. 0 0 0. 0. 0. 0.8 70 Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8. Comparing Linear, Eponential, and Quadratic Functions (continued) EXPLORATION: Comparing Speeds Work with a partner. Analze the speeds of the three cars over the given time periods. The distance traveled in t minutes is miles. Which car eventuall overtakes the others? t = t t t = 1 t = t 1 1 1 3 3 3 5 5 5 7 7 7 8 8 8 9 9 9 Communicate Your Answer 3. How can ou compare the growth rates of linear, eponential, and quadratic functions?. Which function has a growth rate that is eventuall much greater than the growth rates of the other two functions? Eplain our reasoning. Copright Big Ideas Learning, LLC Algebra 1 71
Name Date 8. Notetaking with Vocabular For use after Lesson 8. In our own words, write the meaning of each vocabular term. average rate of change Core Concepts Linear, Eponential, and Quadratic Functions Linear Function Eponential Function Quadratic Function = m + b ab = = a + b + c Notes: Differences and Ratios of Functions You can use patterns between consecutive data pairs to determine which tpe of function models the data. The differences of consecutive -values are called first differences. The differences of consecutive first differences are called second differences. Linear Function The first differences are constant. Eponential Function Consecutive -values have a common ratio. Quadratic Function The second differences are constant. In all cases, the differences of consecutive -values need to be constant. Notes: 7 Algebra 1 Copright Big Ideas Learning, LLC
Name Date 8. Notetaking with Vocabular (continued) Comparing Functions Using Average Rates of Change Over the same interval, the average rate of change of a function increasing quadraticall eventuall eceeds the average rate of change of a function increasing linearl. So, the value of the quadratic function eventuall eceeds the value of the linear function. Over the same interval, the average rate of change of a function increasing eponentiall eventuall eceeds the average rate of change of a function increasing linearl or quadraticall. So, the value of the eponential function eventuall eceeds the value of the linear or quadratic function. Notes: Etra Practice In Eercises 1, plot the points. Tell whether the points appear to represent a linear, an eponential, or a quadratic function. 1. ( 3,, ) (,, ) (,, ) ( 1,8, ) ( 5,8). ( 3,1 ), (, ), ( 1, ), ( 0, 8 ), (,1) 3. (, 0 ), (,1 ), ( 0, 3 ), ( 1, ), (,10). (, ), ( 0, ), (, 0 ), (, ), (, ) Copright Big Ideas Learning, LLC Algebra 1 73
Name Date 8. Notetaking with Vocabular (continued) In Eercises 5 and, tell whether the table of values represents a linear, an eponential, or a quadratic function. 5. 1 0 1 7 1 5. 1 0 1 0 In Eercises 7 and 8, tell whether the data represent a linear, an eponential, or a quadratic function. Then write the function. 7. (,, ) ( 1, 1, ) ( 0,,1,5, ) ( ) (,8) 8. (, 5 ), ( 1,1 ), ( 0, 3 ), ( 1,1 ), (, 5) 9. A ball is dropped from a height of 305 feet. The table shows the height h (in feet) of the ball t seconds after being dropped. Let the time t represent the independent variable. Tell whether the data can be modeled b a linear, an eponential, or a quadratic function. Eplain. Time, t 0 1 3 Height, h 305 89 1 11 9 7 Algebra 1 Copright Big Ideas Learning, LLC