Up and Down or Down and Up

Transcription

1 Lesson.1 Skills Practice Name Date Up and Down or Down and Up Eploring Quadratic Functions Vocabular Write the given quadratic function in standard form. Then describe the shape of the graph and whether it has an absolute maimum or absolute minimum. Eplain our reasoning Problem Set Write each quadratic function in standard form. 1. f() 5 ( 1 3) f() 5 ( 1 3) f() 5? 1? 3 f() f() 5 3( 2 8) g(s) 5 (s 1 4)s d(t) 5 (20 1 3t)t Chapter Skills Practice 581

2 Lesson.1 Skills Practice page 2 5. f(n) 5 2n(3n 2 6) 6. m(s) 5 s(s 1 3) 3 4 Write a quadratic function in standard form that represents each area as a function of the width. Remember to define our variables. 7. A builder is designing a rectangular parking lot. She has 300 feet of fencing to enclose the parking lot around three sides. Let 5 the width of the parking lot Area of a rectangle 5 width 3 length The length of the parking lot A 5 w? l Let A 5 the area of the parking lot A() 5? ( ) 5? 300 2? Aiko is enclosing a new rectangular flower garden with a rabbit garden fence. She has 40 feet of fencing. 9. Pedro is building a rectangular sandbo for the communit park. The materials available limit the perimeter of the sandbo to at most 100 feet. 582 Chapter Skills Practice

3 Lesson.1 Skills Practice page 3 Name Date 10. Lea is designing a rectangular quilt. She has 16 feet of piping to finish the quilt around three sides.. Kiana is making a rectangular vegetable garden alongside her home. She has 24 feet of fencing to enclose the garden around the three open sides. 12. Nelson is building a rectangular ice rink for the communit park. The materials available limit the perimeter of the ice rink to at most 250 feet. Use our graphing calculator to determine the absolute maimum of each function. Describe what the - and -coordinates of this point represent in terms of the problem situation. 13. A builder is designing a rectangular parking lot. He has 400 feet of fencing to enclose the parking lot around three sides. Let 5 the width of the parking lot. Let A 5 the area of the parking lot. The function A() represents the area of the parking lot as a function of the width. The absolute maimum of the function is at (100, 20,000). The -coordinate of 100 represents the width in feet that produces the maimum area. The -coordinate of 20,000 represents the maimum area in square feet of the parking lot. Chapter Skills Practice 583

4 Lesson.1 Skills Practice page Joelle is enclosing a portion of her ard to make a pen for her ferrets. She has 20 feet of fencing. Let 5 the width of the pen. Let A 5 the area of the pen. The function A() represents the area of the pen as a function of the width. 15. A baseball is thrown upward from a height of 5 feet with an initial velocit of 42 feet per second. Let t 5 the time in seconds after the baseball is thrown. Let h 5 the height of the baseball. The quadratic function h(t) t t 1 5 represents the height of the baseball as a function of time. 16. Hector is standing on top of a plaground set at a park. He throws a water balloon upward from a height of 12 feet with an initial velocit of 25 feet per second. Let t 5 the time in seconds after the balloon is thrown. Let h 5 the height of the balloon. The quadratic function h(t) t t 1 12 represents the height of the balloon as a function of time. 17. Franco is building a rectangular roller-skating rink at the communit park. The materials available limit the perimeter of the skating rink to at most 180 feet. Let 5 the width of the skating rink. Let A 5 the area of the skating rink. The function A() represents the area of the skating rink as a function of the width. 18. A football is thrown upward from a height of 6 feet with an initial velocit of 65 feet per second. Let t 5 the time in seconds after the football is thrown. Let h 5 the height of the football. The quadratic function h(t) t t 1 6 represents the height of the football as a function of time. 584 Chapter Skills Practice

5 Lesson.2 Skills Practice Name Date Just U and I Comparing Linear and Quadratic Functions Vocabular Write a definition for each term in our own words. 1. leading coefficient 2. second differences Problem Set Graph each table of values. Describe the tpe of function represented b the graph The function represented b the graph is a linear function Chapter Skills Practice 585

6 Lesson.2 Skills Practice page Chapter Skills Practice

7 Lesson.2 Skills Practice page 3 Name Date Chapter Skills Practice 587

8 Lesson.2 Skills Practice page 4 Calculate the first and second differences for each table of values. Describe the tpe of function represented b the table First Differences Second Differences First Differences Second Differences The function represented b the table is a linear function First Differences Second Differences First Differences Second Differences Chapter Skills Practice

9 Lesson.2 Skills Practice page 5 Name Date First Differences Second Differences First Differences Second Differences Chapter Skills Practice 589

10 590 Chapter Skills Practice

11 Lesson.3 Skills Practice Name Date Walking the... Curve? Domain, Range, Zeros, and Intercepts Vocabular Choose the term that best completes each sentence. zeros vertical motion model interval open interval closed interval half-closed interval half-open interval 1. An is defined as the set of real numbers between two given numbers. 2. The -intercepts of a graph of a quadratic function are also called the of the quadratic function. 3. An (a, b) describes the set of all numbers between a and b, but not including a or b. 4. A or (a, b] describes the set of all numbers between a and b, including b but not including a. Or, [a, b) describes the set of all numbers between a and b, including a but not including b. 5. A quadratic equation that models the height of an object at a given time is a. 6. A [a, b] describes the set of all numbers between a and b, including a and b. Chapter Skills Practice 591

12 Lesson.3 Skills Practice page 2 Problem Set Graph the function that represents each problem situation. Identif the absolute maimum, zeros, and the domain and range of the function in terms of both the graph and problem situation. Round our answers to the nearest hundredth, if necessar. 1. A model rocket is launched from the ground with an initial velocit of 120 feet per second. The function g(t) t t represents the height of the rocket, g(t), t seconds after it was launched. Height (feet) Absolute maimum: (3.75, 225) Zeros: (0, 0), (7.5, 0) 320 Domain of graph: The domain is all real 240 numbers from negative infinit to positive 160 infinit. Domain of the problem: The domain is all 80 real numbers greater than or equal to 0 and less than or equal to Range of graph: The range is all real 2160 numbers less than or equal to Range of the problem: The range is all real numbers less than or equal to 225 and 2320 greater than or equal to 0. Time (seconds) 2. A model rocket is launched from the ground with an initial velocit of 60 feet per second. The function g(t) t t represents the height of the rocket, g(t), t seconds after it was launched Chapter Skills Practice

13 Lesson.3 Skills Practice page 3 Name Date 3. A baseball is thrown into the air from a height of 5 feet with an initial vertical velocit of 15 feet per second. The function g(t) t t 1 5 represents the height of the baseball, g(t), t seconds after it was thrown A football is thrown into the air from a height of 6 feet with an initial vertical velocit of 50 feet per second. The function g(t) t t 1 6 represents the height of the football, g(t), t seconds after it was thrown Chapter Skills Practice 593

14 Lesson.3 Skills Practice page 4 5. A tennis ball is dropped from a height of 25 feet. The initial velocit of an object that is dropped is 0 feet per second. The function g(t) t represents the height of the tennis ball, g(t), t seconds after it was dropped A tennis ball is dropped from a height of 150 feet. The initial velocit of an object that is dropped is 0 feet per second. The function g(t) t represents the height of the tennis ball, g(t), t seconds after it was dropped Chapter Skills Practice

15 Lesson.3 Skills Practice page 5 Name Date Use interval notation to represent each interval described. 7. All real numbers greater than or equal to 23 but less than 5. [23, 5) 8. All real numbers greater than or equal to All real numbers greater than 236 and less than or equal to All real numbers less than or equal to b.. All real numbers greater than or equal to c and less than or equal to d. 12. All real numbers greater than or equal to n. Identif the intervals of increase and decrease for each function. 13. f() f() Interval of increase: (23, `) Interval of decrease: (2`, 23) Chapter Skills Practice 595

16 Lesson.3 Skills Practice page f() f() f() f() Chapter Skills Practice

17 Lesson.4 Skills Practice Name Date Are You Afraid of Ghosts? Factored Form of a Quadratic Function Vocabular Write a definition for each term in our own words. 1. factor an epression 2. factored form Problem Set Factor each epression () 2 6(4) 5 6( 2 4) Chapter Skills Practice 597

18 Lesson.4 Skills Practice page 2 Determine the -intercepts of each quadratic function in factored form. 7. f() 5 ( 2 2)( 2 8) The -intercepts are (2, 0) and (8, 0). 8. f() 5 ( 1 1)( 2 6) 9. f() 5 3( 1 4)( 2 2) 10. f() ( 2 1)( 2 12). f() 5 0.5( 1 15)( 1 5) 12. f() 5 4( 2 1)( 2 9) Write a quadratic function in factored form with each set of given characteristics. 13. Write a quadratic function that represents a parabola that opens downward and has -intercepts (22, 0) and (5, 0). Answers will var but functions should be in the form: f() 5 a( 1 2)( 2 5) for a, Write a quadratic function that represents a parabola that opens downward and has -intercepts (2, 0) and (14, 0). 15. Write a quadratic function that represents a parabola that opens upward and has -intercepts (28, 0) and (21, 0). 16. Write a quadratic function that represents a parabola that opens upward and has -intercepts (3, 0) and (7, 0). 598 Chapter Skills Practice

19 Lesson.4 Skills Practice page 3 Name Date 17. Write a quadratic function that represents a parabola that opens downward and has -intercepts (25, 0) and (2, 0). 18. Write a quadratic function that represents a parabola that opens upward and has -intercepts (212, 0) and (24, 0). Determine the -intercepts for each function using our graphing calculator. Write the function in factored form. 19. f() intercepts: (1, 0) and (7, 0) factored form: f() 5 ( 2 1)( 2 7) 20. f() f() f() f() f() Chapter Skills Practice 599

20 Lesson.4 Skills Practice page 4 Determine the -intercepts for each function. If necessar, rewrite the function in factored form. 25. f() 5 (3 1 18)( 2 2) factored form: f() 5 3( 1 6)( 2 2) -intercepts: (26, 0) and (2, 0) 26. f() 5 ( 1 8)(3 2 ) 27. f() 5 (22 1 8)( 2 14) 28. f() 5 ( 1 16)(2 1 16) 29. f() 5 ( 1 7) 30. f() 5 (23 1 9)( 1 3) 600 Chapter Skills Practice

21 Lesson.5 Skills Practice Name Date Just Watch That Pumpkin Fl! Investigating the Verte of a Quadratic Function Vocabular Graph the quadratic function. Plot and label the verte. Then draw and label the ais of smmetr. Eplain how ou determine each location. h(t) 5 t 2 1 2t Problem Set Write a function that represents the vertical motion described in each problem situation. 1. A catapult hurls a watermelon from a height of 36 feet at an initial velocit of 82 feet per second. h(t) t 2 1 v 0 t 1 h 0 h(t) t t A catapult hurls a cantaloupe from a height of 12 feet at an initial velocit of 47 feet per second. Chapter Skills Practice 601

22 Lesson.5 Skills Practice page 2 3. A catapult hurls a pineapple from a height of 49 feet at an initial velocit of 0 feet per second. 4. A basketball is thrown from a height of 7 feet at an initial velocit of 58 feet per second. 5. A soccer ball is thrown from a height of 25 feet at an initial velocit of 46 feet per second. 6. A football is thrown from a height of 6 feet at an initial velocit of 74 feet per second. Identif the verte and the equation of the ais of smmetr for each vertical motion model. 7. A catapult hurls a grapefruit from a height of 24 feet at an initial velocit of 80 feet per second. The function h(t) t t 1 24 represents the height of the grapefruit h(t) in terms of time t. The verte of the graph is (2.5, 124). The ais of smmetr is A catapult hurls a pumpkin from a height of 32 feet at an initial velocit of 96 feet per second. The function h(t) t t 1 32 represents the height of the pumpkin h(t) in terms of time t. 9. A catapult hurls a watermelon from a height of 40 feet at an initial velocit of 64 feet per second. The function h(t) t t 1 40 represents the height of the watermelon h(t) in terms of time t. 10. A baseball is thrown from a height of 6 feet at an initial velocit of 32 feet per second. The function h(t) t t 1 6 represents the height of the baseball h(t) in terms of time t. 602 Chapter Skills Practice

23 Lesson.5 Skills Practice page 3 Name Date. A softball is thrown from a height of 20 feet at an initial velocit of 48 feet per second. The function h(t) t t 1 20 represents the height of the softball h(t) in terms of time t. 12. A rocket is launched from the ground at an initial velocit of 2 feet per second. The function h(t) t 2 1 2t represents the height of the rocket h(t) in terms of time t. Determine the ais of smmetr of each parabola. 13. The -intercepts of a parabola are (3, 0) and (9, 0) The ais of smmetr is The -intercepts of a parabola are (23, 0) and (1, 0). 15. The -intercepts of a parabola are (212, 0) and (22, 0). 16. Two smmetric points on a parabola are (21, 4) and (5, 4). 17. Two smmetric points on a parabola are (24, 8) and (2, 8). Chapter Skills Practice 603

24 Lesson.5 Skills Practice page Two smmetric points on a parabola are (3, 1) and (15, 1). Determine the verte of each parabola. 19. f() ais of smmetr: 5 21 The ais of smmetr is The -coordinate of the verte is 21. The -coordinate when 5 21 is: f(21) 5 (21 ) 2 1 2(21) The verte is (21, 216). 20. f() ais of smmetr: f() intercepts: (2, 0) and (26, 0) 22. f() intercepts: (29, 0) and (25, 0) 23. f() two smmetric points on the parabola: (21, ) and (9, ) 24. f() two smmetric points on the parabola: (23, 7) and (3, 7) 604 Chapter Skills Practice

25 Lesson.5 Skills Practice page 5 Name Date Determine another point on each parabola. 25. The ais of smmetr is 5 3. A point on the parabola is (1, 4). Another point on the parabola is a smmetric point that has the same -coordinate as (1, 4). The -coordinate is: 1 1 a a 5 6 a 5 5 Another point on the parabola is (5, 4). 26. The ais of smmetr is A point on the parabola is (0, 6). 27. The ais of smmetr is 5 1. A point on the parabola is (23, 2). 28. The verte is (5, 2). A point on the parabola is (3, 21). 29. The verte is (21, 6). A point on the parabola is (2, 3). 30. The verte is (3, 21). A point on the parabola is (4, 1). Chapter Skills Practice 605

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27 Lesson.6 Skills Practice Name Date The Form Is Ke Verte Form of a Quadratic Function Vocabular Write a definition for the term in our own words. 1. verte form Problem Set Determine the verte of each quadratic function given in verte form. 1. f() 5 ( 2 3) The verte is (3, 8). 2. f() 5 ( 1 4) f() 5 22( 2 1) f() ( 2 2) f() 5 2( 1 9) f() 5 ( 2 5) 2 Determine the verte of each quadratic function given in standard form. Use our graphing calculator. Rewrite the function in verte form. 7. f() The verte is (3, 236). The function in verte form is f() 5 ( 2 3) f() f() f() Chapter Skills Practice 607

28 Lesson.6 Skills Practice page 2. f() f() Determine the -intercepts of each quadratic function given in standard form. Use our graphing calculator. Rewrite the function in factored form. 13. f() The -intercepts are (2, 0) and (24, 0). The function in factored form is f() 5 ( 2 2)( 1 4). 14. f() f() f() f() f() Identif the form of each quadratic function as either standard form, factored form, or verte form. Then state all ou know about the quadratic function s ke characteristics, based onl on the given equation of the function. 19. f() 5 5( 2 3) The function is in verte form. The parabola opens up and the verte is (3, 12). 20. f() 5 2( 2 8)( 2 4) 21. f() f() 5 2 ( 1 6)( 2 1) f() 5 2( 1 2) f() Chapter Skills Practice

29 Lesson.6 Skills Practice page 3 Name Date Write an equation for a quadratic function with each set of given characteristics. 25. The verte is (21, 4) and the parabola opens down. Answers will var but functions should be in the form: f() 5 a( 2 h) 2 1 k f() 5 a( 1 1) 2 1 4, for a, The -intercepts are 23 and 4 and the parabola opens down. 27. The verte is (3, 22) and the parabola opens up. 28. The verte is (0, 8) and the parabola opens up. 29. The -intercepts are 5 and 12 and the parabola opens up. 30. The -intercepts are 0 and 7 and the parabola opens down. Chapter Skills Practice 609

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31 Lesson.7 Skills Practice Name Date More Than Meets the Ee Transformations of Quadratic Functions Vocabular Write a definition for each term in our own words. 1. vertical dilation 2. dilation factor Problem Set Describe the transformation performed on each function g() to result in d(). 1. g() 5 2 d() The graph of g() is translated down 5 units. 2. g() 5 2 d() g() d() g() 5 ( 1 2) 2 d() 5 ( 1 2) g() d() g() 5 2( 2 2) 2 d() 5 2( 2 2) Chapter Skills Practice 6

32 Lesson.7 Skills Practice page 2 Describe the transformation performed on each function g() to result in m(). 7. g() 5 2 m() 5 ( 1 4) 2 The graph of g() is translated left 4 units. 8. g() 5 2 m() 5 ( 2 8) 2 9. g() g() m() 5 ( 1 1) 2 m() 5 ( 1 2) g() m() 5 ( 1 3) g() m() 5 ( 2 5) Describe the transformation performed on each function g() to result in p(). 13. g() 5 2 p() The graph of p() is a horizontal reflection of the graph of g(). 15. g() p() 5 2( 2 1 2) 14. g() 5 2 p() 5 (2) g() p() 5 (2) g() p() (2) g() p() 5 2( ) Represent each function n() as a vertical dilation of g() using coordinate notation. 19. g() 5 2 n() (, ) (, 4) 20. g() 5 2 n() Chapter Skills Practice

33 Lesson.7 Skills Practice page 3 Name Date 21. g() n() g() 5 22 n() g() 5 ( 1 1) 2 n() 5 2( 1 1) g() 5 ( 2 3)2 n() 5 1 ( 2 3)2 2 Write an equation in verte form for a function g() with the given characteristics. Sketch a graph of each function g(). 25. The function g() is quadratic. The function g() is continuous. The graph of g() is a horizontal reflection of the graph of f() 5 2. The function g() is translated 3 units up from f() g() 5 2( 2 0) Chapter Skills Practice 613

34 Lesson.7 Skills Practice page The function g() is quadratic. The function g() is continuous. The graph of g() is a horizontal reflection of the graph of f() 5 2. The function g() is translated 2 units down and 5 units left from f() The function g() is quadratic. The function g() is continuous. The function g() is verticall dilated with a dilation factor of 6. The function g() is translated 1 unit up and 4 units right from f() Chapter Skills Practice

35 Lesson.7 Skills Practice page 5 Name Date 28. The function g() is quadratic. The function g() is continuous. The function g() is verticall dilated with a dilation factor of 1 2. The function g() is translated 2 units down and 6 units left from f() The function g() is quadratic. The function g() is continuous. The graph of g() is a horizontal reflection of the graph of f() 5 2. The function g() is verticall dilated with a dilation factor of 3. The function g() is translated 2 units down and 4 units right from f() Chapter Skills Practice 615

36 Lesson.7 Skills Practice page The function g() is quadratic. The function g() is continuous. The function g() is verticall dilated with a dilation factor of 1 4. The function g() is translated 3 units up and 2 units left from f() Describe the transformation(s) necessar to translate the graph of the function f() 5 2 into the graph of each function g(). 31. g() The function g() is translated 7 units up from f() g() g() 5 ( 2 2) g() g() ( 1 4) g() 5 2( 2 6) Chapter Skills Practice