Shape and Structure. Forms of Quadratic Functions. Lesson 4.1 Skills Practice. Vocabulary

Transcription

1 Lesson.1 Skills Practice Name Date Shape and Structure Forms of Quadratic Functions Vocabular Write an eample for each form of quadratic function and tell whether the form helps determine the -intercepts, the -intercept, or the verte of the graph. Then describe how to determine the concavit of a parabola. 1. Standard form:. Factored form: 3. Verte form:. Concavit of a parabola: Chapter Skills Practice 339

2 Lesson.1 Skills Practice page Problem Set Circle the function that matches each graph. Eplain our reasoning. 1.. f() 5 ( )( ) f() 5 1 ( 1 )( 1 ) f() 5 1 ( 1 )( 1 ) f() 5 1 ( )( ) The a value is positive so the parabola opens up. Also, the roots are at and. f() f() f() f() f() 5.5( ) 1 f() 5 3( 1 )( 5) f() 5 ( ) f() 5.5( 1 ) 1 f() 5.5( ) 1 f() 5 3( 1 )( 1 5) f() 5 3( )( 5) f() 5 3( )( 5) 3 Chapter Skills Practice

3 Lesson.1 Skills Practice page 3 Name Date f() f() f() f() f() 5 1 ( ) f() 5 1 ( ) 1 f() 5 1 ( ) f() 5 1 ( 1 ) Use the given information to determine the most efficient form ou could use to write the quadratic function. Write standard form, factored form, or verte form. 7. verte (3, 7) and point (1, 1) verte form. points (1, ), (, 3), and (7, ) 9. -intercept (, 3) and ais of smmetr 3 1. points (1, 1), (5, 1), and (, ) 11. roots (5, ), (13, ) and point (7, ) 1. maimum point (, ) and point (3, 15) Chapter Skills Practice 31

4 Lesson.1 Skills Practice page Convert each quadratic function in factored form to standard form. 13. f() 5 ( 1 5)( 7) 1. f() 5 ( 1 )( 1 9) f() f() 5 ( )( 1 1) 1. f() 5 3( 1)( 3) 17. f() 5 1 ( 1 )( 1 3) 1. f() 5 5 ( )( 1 ) 3 Convert each quadratic function in verte form to standard form. 19. f() 5 3( ) 1 7. f() 5 ( 1 1) 5 f() 5 3( 1 1) f() 5 ( 1 7 ) 3. f() 5 ( ) 1 3. f() 5 1 ( 1) 1. f() 5 1 ( 1 1) 1 3 Chapter Skills Practice

5 Lesson.1 Skills Practice page 5 Name Date Write a quadratic function to represent each situation using the given information. 5. Cor is training his dog, Cocoa, for an agilit competition. Cocoa must jump through a hoop in the middle of a course. The center of the hoop is feet from the starting pole. The dog runs from the starting pole for 5 feet, jumps through the hoop, and lands feet from the hoop. When Cocoa is 1 foot from landing, Cor measures that she is 3 feet off the ground. Write a function to represent Cocoa s height in terms of her distance from the starting pole. h(d) 5 a(d r 1 ) (d r ) 3 5 a(11 5) (11 1) 3 5 a()(1) 3 5 a 3 5 a.5 5 a h(d) 5.5(d 5) (d 1). Sasha is training her dog, Bingo, to run across an arched ramp, which is in the shape of a parabola. To help Bingo get across the ramp, Sasha places a treat on the ground where the arched ramp begins and one at the top of the ramp. The treat at the top of the ramp is a horizontal distance of feet from the first treat, and Bingo is feet above the ground when he reaches the top of the ramp. Write a function to represent Bingo s height above the ground as he walks across the ramp in terms of his distance from the beginning of the ramp. Chapter Skills Practice 33

6 Lesson.1 Skills Practice page 7. Ella s dog, Doug, is performing in a special tricks show. Doug can fling a ball off his nose into a bucket feet awa. Ella places the ball on Doug s nose, which is feet off the ground. Doug flings the ball through the air into a bucket sitting on a -foot platform. Halfwa to the bucket, the ball is 1 feet in the air. Write a function to represent the height of the ball in terms of its distance from Doug.. A spectator in the crowd throws a treat to one of the dogs in a competition. The spectator throws the treat from the bleachers 19 feet above ground. The treat amazingl flies 3 feet and just barel crosses over a hoop which is 7.5 feet tall. The dog catches the treat feet beond the hoop when his mouth is 1 foot from the ground. Write a function to represent the height of the treat in terms of its distance. 3 Chapter Skills Practice

7 Lesson.1 Skills Practice page 7 Name Date 9. Hector s dog, Ginger, competes in a waterfowl jump. She jumps from the edge of the water, catches a to duck at a horizontal distance of 1 feet from the edge of the water and a height of feet above the water, and lands in the water at a horizontal distance of 15 feet from the edge of the water. Write a function to represent the height of Ginger s jump in terms of her horizontal distance. 3. Ping is training her dog, TinTin, to jump across a row of logs. He takes off from a platform that is 7 feet high with a speed of 1 feet per second. Write a function to represent TinTin s height in terms of time as he jumps across the logs. Chapter Skills Practice 35

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9 Lesson. Skills Practice Name Date Function Sense Translating Functions Vocabular Complete each sentence with the correct term from the word bank. transformation translation reference point argument of a function 1. A(n) is one of a set of ke points that help identif the basic function.. The mapping, or movement, of all the points of a figure in a plane according to a common operation is called a(n). 3. The is the variable, term, or epression on which the function operates.. A(n) is a tpe of transformation that shifts an entire figure or graph the same distance and direction. Problem Set Given f() 5, complete the table and graph h(). 1. h() 5 ( 1) 1 3 Reference Points on f() S Corresponding Points on h() h() (, ) S (1, 3) (1, 1) S (, ) (, ) S (3, 7) Chapter Skills Practice 37

10 Lesson. Skills Practice page. h() 5 ( 1 ) 1 Reference Points of f() S Corresponding Points on h() (, ) S (1, 1) S (, ) S 3. h() 5 ( 1 7) Reference Points of f() S Corresponding Points on h() (, ) S (1, 1) S (, ) S 1 3 Chapter Skills Practice

11 Lesson. Skills Practice page 3 Name Date. h() 5 ( 3) 1 Reference Points of f() S Corresponding Points on h() (, ) S (1, 1) S (, ) S 5. h() 5 9 Reference Points of f() S Corresponding Points on h() (, ) S (1, 1) S (, ) S 1 Chapter Skills Practice 39

12 Lesson. Skills Practice page. h() 5 ( 1 ) Reference Points of f() S Corresponding Points on h() (, ) S (1, 1) S (, ) S Each given function is in transformational function form g() 5 Af(B( C)) 1 D, where f() 5. Identif the values of C and D for the given function. Then, describe how the verte of the given function compares to the verte of f(). 7. g() 5 f( ) 1 1 The C value is and the D value is 1, so the verte will be shifted units to the right and 1 units up to (, 1).. g() 5 f( 1 ) 9 9. g() 5 f( 5) g() 5 f( ) Chapter Skills Practice

13 Lesson. Skills Practice page 5 Name Date 11. g() 5 f( 1 ) g() 5 f( 1 ) Analze the graphs of b(), c(), d(), and f(). Write each function in terms of the indicated function. b() d() f() c() 13. Write b() in terms of f(). b() 5 f( 1 5) 1. Write c() in terms of f(). 15. Write d() in terms of f(). 1. Write d() in terms of b(). 17. Write c() in terms of b(). 1. Write b() in terms of c(). Chapter Skills Practice 351

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15 Lesson.3 Skills Practice Name Date Up and Down Vertical Dilations of Quadratic Functions Vocabular 1. Label the graph to identif the vertical dilations (vertical compression and vertical stretching) and the reflection of the function f() 5. Also label the line of reflection. f() 5 Problem Set Graph each vertical dilation of f() 5 and tell whether the transformation is a vertical stretch or a vertical compression and if the graph includes a reflection. 1. g() 5. p() 5 1 f() f() vertical stretch Chapter Skills Practice 353

16 Lesson.3 Skills Practice page 3. h() 5 5. m() f() f() d() f(). g() f() Chapter Skills Practice

17 Lesson.3 Skills Practice page 3 Name Date Each given function is in transformational function form g() 5 Af(B( C)) 1 D, where f() 5. Describe how g() compares to f(). Then, use coordinate notation to represent how the A-, C-, and D-values transform f() to generate g(). 7. g() 5 3(f()) 1 The A value is 3, so the graph will have a vertical stretch b a factor of 3 and will be reflected about the line 5 1. The C value is and the D value is 1 so the verte will be shifted 1 unit down to (, 1). (, ) (, 3 1). g() 5 1 (f()) 1 9. g() 5 (f( 1 3)) 1. g() 5 1 f( ) g() 5.75f( 1 ) 1. g() 5 3 f ( 1 3 ) 1 3 Chapter Skills Practice 355

18 Lesson.3 Skills Practice page Write the function that represents each graph f() 5 3( 1 ) Chapter Skills Practice

19 Lesson. Skills Practice Name Date Side to Side Horizontal Dilations of Quadratic Functions Vocabular 1. Eplain the differences and similarities between horizontal dilation, horizontal stretching, and horizontal compression of a quadratic function. Problem Set Complete the table and graph m(). Then, describe how the graph of m() compares to the graph of f(). 1. f() 5 ; m() 5 f ( 1 5 ) Reference Points on f() S Corresponding Points on m() (, ) S (, ) (5, 5) S (5, 1) (1, 1) S (1, ) (15, 5) S (15, 9) The function m() is a horizontal stretch of f() b a factor of m() Chapter Skills Practice 357

20 Lesson. Skills Practice page. f() 5 ; m() 5 f(1.5) Reference Points on f() S Corresponding Points on m() (, ) S (1, 1) S (, ) S (, 1) S 3. f() 5 ; m() 5 f() Reference Points on f() S Corresponding Points on m() 1 (, ) S (.5,.5) S (1, 1) S (, ) S f() 5 ; m() 5 f(.5) Reference Points on f() S (, ) S (, 1) S (, ) S Corresponding Points on m() (1, 1) S 35 Chapter Skills Practice

21 Lesson. Skills Practice page 3 Name Date 5. f() 5 ; m() 5 f ( 3 ) Reference Points on f() S (, ) S (3, 9) S (, 3) S (9, 1) S Corresponding Points on m() f() 5 ; m() 5 f () Reference Points on f() S (, ) S (1, 1) S (, ) S (3, 9) S Corresponding Points on m() Chapter Skills Practice 359

22 Lesson. Skills Practice page The graph of f() is shown. Sketch the graph of the given transformed function. 7. d() 5 f(). t() 5 f( ) d() f() f() 9. m() 5 f( 1 3) g() 5 ( 1 1) f() f() 3 Chapter Skills Practice

23 Lesson. Skills Practice page 5 Name Date 11. r() 5 f ( ) 1 1. p() 5 f( 1 1) 3 f() 1 f() Write an equation for w() in terms of v() v() w() w() v() w() 5 v ( 1 ) Chapter Skills Practice 31

24 Lesson. Skills Practice page w() v() v() w() v() w() w() 3 v() 3 Chapter Skills Practice

25 Lesson.5 Skills Practice Name Date What s the Point? Deriving Quadratic Functions Problem Set Use our knowledge of reference points to write an equation for the quadratic function that satisfies the given information. Use the graph to help solve each problem. 1. Given: verte (3, 5) and point (5, 3) f() 5 ( 3) 1 5 Point (5, 3) is point B because it is units from the ais of smmetr. The range between the verte and point B on the basic function is. The range between the verte and point B is 3 (), therefore the a-value must be. (3, 5) (5, 3) B units. Given: verte (, 9) and one of two -intercepts (1, ) (1, ) (, 9) Chapter Skills Practice 33

26 Lesson.5 Skills Practice page 3. Given: two -intercepts (7, ) and (5, ) and one point (, 9) (7, ) (5, ) (, 9). Given: verte (, 3) and -intercept (, 11) 1 1 (, 11) (, 3) 3 Chapter Skills Practice

27 Lesson.5 Skills Practice page 3 Name Date 5. Given: eactl one -intercept (, ) and -intercept (, 1) (, ) 1 (, 1). Given: verte (, 1) and point (3, 35) (3, 35) 3 1 (, 1) 1 3 Chapter Skills Practice 35

28 Lesson.5 Skills Practice page Use a graphing calculator to determine the quadratic equation for each set of three points that lie on a parabola. 7. (, 1), (, 1), (, ) f() = (5, 5), (1, ), (1, ) 9. (, ), (, ), (, 3) 1. (, 3), (, 9), (5, ) 11. (, 3), (5,. ), (15, 7.) 1. (, 13), (1, 17), (7, 31) 3 Chapter Skills Practice

29 Lesson.5 Skills Practice page 5 Name Date Create a sstem of equations and use algebra to write a quadratic equation for each set of three points that lie on a parabola. 13. (3, 1), (, 9), (3, ) Equation 1: 1 5 9a 3b 1 c Equation : 9 5 c Equation 3: 5 9a 1 3b 1 c Substitute equation into equation 1 and solve for a a 3b a 3b 3 1 3b 5 9a 1 a b Substitute the value for a in terms of b and the value for c into equation 3 and solve for b. 5 9 ( b ) 1 3b b 1 3b b 1 5 b b 5 Substitute the values for b and c into equation 1 and solve for a a 3() a a a 5 1 Substitute the values for a, b, and c into a quadratic equation in standard form. f() Chapter Skills Practice 37

30 Lesson.5 Skills Practice page 1. (, ), (1, 5), (, 1) 3 Chapter Skills Practice

31 Lesson.5 Skills Practice page 7 Name Date 15. (, 9), (, 5), (1, 15) Chapter Skills Practice 39

32 Lesson.5 Skills Practice page 1. (1, ), (, 7), (3, ) 37 Chapter Skills Practice

33 Lesson.5 Skills Practice page 9 Name Date 17. (5, ), (, ), (3, ) Chapter Skills Practice 371

34 Lesson.5 Skills Practice page 1 1. (1, 17), (1, 9), (, 15) 37 Chapter Skills Practice

35 Lesson. Skills Practice Name Date Now It s Getting Comple... But It s Reall Not Difficult! Comple Number Operations Vocabular Match each term to its corresponding definition. 1. the number i A. a number in the form a 1 bi where a and b are real numbers and b is not equal to. imaginar number B. term a of a number written in the form a 1 bi 3. pure imaginar number C. a polnomial with two terms. comple number D. pairs of numbers of the form a 1 bi and a bi 5. real part of a comple number E. a number such that its square equals 1. imaginar part of a comple number F. a number in the form a 1 bi where a and b are real numbers 7. comple conjugates G. a polnomial with three terms. monomial H. a number of the form bi where b is not equal to 9. binomial I. term bi of a number written in the form a 1 bi 1. trinomial J. a polnomial with one term Chapter Skills Practice 373

36 Lesson. Skills Practice page Problem Set Calculate each power of i. 1. i i 5 (i ) i i 55. i 1 5. i. i 7 37 Chapter Skills Practice

37 Lesson. Skills Practice page 3 Name Date Rewrite each epression using i ()(1) 5 i Chapter Skills Practice 375

38 Lesson. Skills Practice page Simplif each epression. 15. ( 1 5i ) (7 9i ) ( 1 5i ) (7 9i ) 5 1 5i 7 1 9i 5 ( 7) 1 (5i 1 9i ) i 1. 1 i 1 11i (i 1 1 3i ) 1 (i ) 1. i 1 13 (7i i ) 1 1i i(7 i ). ( 5i )( 1 i ) 1..5(1i ) i(.75 3i ). ( 1 i 3 ) ( 1 3 i ) 37 Chapter Skills Practice

39 Lesson. Skills Practice page 5 Name Date Determine each product. 3. (3 1 i )(3 i ) (3 1 i )(3 i ) 5 9 3i 1 3i i 5 9 (1) 5 1. (i 5)(i 1 5) 5. (7 i )(7 1 i ). ( i ) ( 1 3 3i ) 7. (.1 1.i )(.1.i ). [(i )(i 1 )] Identif each epression as a monomial, binomial, or trinomial. Eplain our reasoning. 9. i 1 7 The epression is a monomial because it can be rewritten as (i 1 7), which shows one term i i 1 3 Chapter Skills Practice 377

40 Lesson. Skills Practice page 3. i i 33. i 1 i 1 i i i 1 Simplif each epression, if possible. 35. ( i ) ( i ) 5 i i 1 3i 5 1i 1 3(1) 5 1i 3 3. ( 1 5i )(7 i ) 37. 3i i 3. (i 9)(3 1 5i ) 37 Chapter Skills Practice

41 Lesson. Skills Practice page 7 Name Date 39. ( 1 i )( i )( 1 i ). (3i i )(3i i ) 1 (i 3i )( 3i ) For each comple number, write its conjugate i 7 i i 3. i. 7i 5. 11i. 9 i i. 1 1 i Chapter Skills Practice 379

42 Lesson. Skills Practice page Calculate each quotient i 5 1 i 3 1 i 5 1 i i 5 1 i? 5 i i 1 i i 5 i 5 3i 1 3i 3i i i i i 1 i i 3i i 1 i 53. 3i i 5. i 1 1 i 3 Chapter Skills Practice

43 Lesson.7 Skills Practice Name Date You Can t Spell Fundamental Theorem of Algebra without F-U-N! Quadratics and Comple Numbers Vocabular Write a definition for each term in our own words. 1. imaginar roots. discriminant 3. imaginar zeros. degree of a polnomial equation 5. Fundamental Theorem of Algebra. double root Chapter Skills Practice 31

44 Lesson.7 Skills Practice page Problem Set Use the Quadratic Formula to solve an equation of the form f() 5 for each function. 1. f() a 5 1, b 5, c 5 3 b b 5 ac a () () (1)(3) 5 (1) f() , f() 5 9. f() Chapter Skills Practice

45 Lesson.7 Skills Practice page 3 Name Date 5. f() f() Use the discriminant to determine whether each function has real or imaginar zeros. 7. f() b ac 5 1 (1)(35) The discriminant is positive, so the function has real zeros.. f() Chapter Skills Practice 33

46 Lesson.7 Skills Practice page 9. f() f() f() f() Chapter Skills Practice

47 Lesson.7 Skills Practice page 5 Name Date Use the verte form of a quadratic equation to determine whether the zeros of each function are real or imaginar. Eplain how ou know. 13. f() 5 ( ) Because the verte (, ) is below the -ais and the parabola is concave up (a. ), it intersects the -ais. So, the zeros are real. 1. f() 5 ( 1) f() ( ) f() 5 3( 1) f() 5 ( ) 1. f() 5 3 ( 1 ) Chapter Skills Practice 35

48 Lesson.7 Skills Practice page Factor each function over the set of real or imaginar numbers. Then, identif the tpe of zeros. 19. k() 5 5 k() 5 ( 1 5)( 5) 5 5, 5 5 The function k() has two real zeros.. n() p() g() h() m() Chapter Skills Practice