Name: ________________________ Class: ___________________ Date: __________

ID: A

1 HW Unit 5.1 Attributes of Cubics Planting the Seeds Exploring Cubic Functions 1. Write a cubic function with zeros of 4, 2, and 3. Write the function in the form f(x)  ax 3  bx 2  cx  d . Verify graphically that the function has the correct zeros. 2. Consider the given functions.

f(x)  x  2 • g(x)  x 2  3.5x  2.5 • h(x)  f(x)  g(x)  (x  2)(x 2  3.5x  2.5) •

a.

Determine the zeros of f(x), g(x), and h(x).

b.

How are the zeros of h(x) related to the zeros of f(x) and g(x). Explain why this is true.

c.

Write a function m(x) that has the same zeros as h(x) plus an additional zero of 5. Verify your answer graphically.

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Name: ________________________

ID: A

Function Makeover Transformations and Symmetry of Polynomial Functions 3. Analyze the graphs of the functions f(x) and g(x).

a.

Write the equation for f(x).

b.

The function g(x) is a transformation of the function f(x). Describe the transformations performed on f(x) that resulted in the function g(x). Explain your reasoning.

c.

Write the equation for g(x).

d.

Is the function g(x) even, odd, or neither? Explain your reasoning.

Determine the product of three linear factors. Verify graphically that the expressions are equivalent. 4. 3x(x  3)(x  2) 5. (2x  1)(2x  1)(x  4) 6. (4x  7) 3 Determine the product of linear and quadratic factors. Verify graphically that the expressions are equivalent. 7. (x 2  1)(8  x)

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Name: ________________________

ID: A

Problem Set Use reference points and symmetry to complete the table of values for each function. Then, graph the function on the coordinate plane. State whether the function is odd, even, or neither. 8. c(x)  x 3 ; h(x)  3c(x) Reference Points on c(x)



(0, 0)



(1, 1)



(2, 8)



Corresponding Points on h(x)

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Name: ________________________

ID: A

9. c(x)  x 3 ; d(x)  c(x)  3 Reference Points on c(x)



(0, 0)



(1, 1)



(2, 8)



Corresponding Points on d(x)

Analyze the graphs of f(x) and g(x). Write an equation for g(x) in terms of f(x). 10. g(x) 

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Name: ________________________

ID: A

11. g(x) 

12. g(x) 

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ID: A 1 HW Unit 5.1 Attributes of Cubics Answer Section 1. ANS: Answers will vary. The function must have factors of (x  4), (x  2), and (x  3).

f(x)  (x  4)(x  2)(x  3) f(x)  (x 2  2x  8)(x  3) f(x)  x 3  3x 2  2x 2  6x  8x  24 f(x)  x 3  x 2  14x  24 The graph of f(x)  x 3  x 2  14x  24 crosses the x-axis at (4, 0), (2, 0), and (3, 0). Therefore, the function has zeros of 4, 2, and 3. PTS: 1 REF: 4.1 STA: 2.A | 6.A | 7.I TOP: Assignment KEY: relative maximum | relative minimum | cubic function | multiplicity 2. ANS: a. By graphing each function, I can determine that f(x) has a zero of 2, g(x) has zeros of 1 and 2.5, and h(x) has zeros of 2, 1, and 2.5. b.

The zeros of h(x) are the combined zeros of f(x) and g(x). This is true because the factors of h(x) are the single factor of f(x) and the two factors of g(x). The factors of a function are directly related to the zeros.

c.

Answers will vary.

m(x)  (x  5)(x  2)(x 2  3.5x  2.5) By graphing, I see that the function m(x) has the same zeros as h(x) plus an additional zero of 5, because it crosses the x-axis at (2, 0), (1, 0), (2.5, 0), and (5, 0). PTS: 1 TOP: Assignment

REF: 4.1 STA: 2.A | 6.A | 7.I KEY: relative maximum | relative minimum | cubic function | multiplicity

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ID: A 3. ANS: a.

f(x)  x 3

b.

The function f(x) was first reflected about the x-axis because reference point A is below the point of symmetry on f(x) and reference point A is above the point of symmetry on g(x). Reference point B in the basic function f(x) is 1 unit below the point of symmetry. The corresponding reference point B  is 0.5 unit above the point of symmetry on g(x). Therefore, the basic function f(x) was vertically dilated by a factor of 0.5. The point of symmetry on g(x) is 3 units above the point of symmetry on f(x). Therefore, the basic function f(x) was vertically translated 3 units up.

c.

g(x)  0.5x 3  3

d.

Neither. The function g(x) is not even because it is not symmetric about the y-axis. The function g(x) is not odd because it is not symmetric about the origin.

PTS: 1 REF: 4.3 STA: 2.A | 6.A KEY: polynomial function | quartic function | quintic function 4. ANS: 3x(x  3)(x  2)  3x(x 2  2x  3x  6)

TOP: Assignment

 3x(x 2  x  6)  3x 3  3x 2  18x The graph of the original expression and the graph of the final expression are the same. So the expressions are equivalent. PTS: 1 REF: 4.1 STA: 2.A | 6.A | 7.I TOP: Skills Practice KEY: relative maximum | relative minimum | cubic function | multiplicity 5. ANS: (2x  1)(2x  1)(x  4)  (4x 2  2x  2x  1)(x  4)

 (4x 2  1)(x  4)  4x 3  16x 2  x  4 The graph of the original expression and the graph of the final expression are the same. So the expressions are equivalent. PTS: 1 REF: 4.1 STA: 2.A | 6.A | 7.I TOP: Skills Practice KEY: relative maximum | relative minimum | cubic function | multiplicity

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ID: A 6. ANS: (4x  7) 3  (4x  7)(4x  7)(4x  7)

 (16x 2  28x  28x  49)(4x  7)  (16x 2  56x  49)(4x  7)  64x 3  112x 2  224x 2  392x  196x  343  64x 3  336x 2  588x  343 The graph of the original expression and the graph of the final expression are the same. So the expressions are equivalent. PTS: 1 REF: 4.1 STA: 2.A | 6.A | 7.I TOP: Skills Practice KEY: relative maximum | relative minimum | cubic function | multiplicity 7. ANS: (x 2  1)(8  x)  8x 2  x 3  8  x

 x 3  8x 2  x  8 The graph of the original expression and the graph of the final expression are the same. So the expressions are equivalent. PTS: 1 REF: 4.1 STA: 2.A | 6.A | 7.I TOP: Skills Practice KEY: relative maximum | relative minimum | cubic function | multiplicity

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ID: A 8. ANS: Reference Points on c(x)



Corresponding Points on h(x)

(0, 0)



(0, 0)

(1, 1)



(1,  3)

(2, 8)



(2,  24)

The function h(x) is an odd function.

PTS: 1 REF: 4.3 STA: 2.A | 6.A KEY: polynomial function | quartic function | quintic function

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TOP: Skills Practice

ID: A 9. ANS: Reference Points on c(x)



(0, 0)



Corresponding Points on d(x) (0,  3)

(1, 1)



(1,  2)

(2, 8)



(2, 5)

The function d(x) is neither even nor odd.

PTS: 1 REF: 4.3 STA: 2.A | 6.A KEY: polynomial function | quartic function | quintic function 10. ANS: g (x)  f (x)  2

TOP: Skills Practice

PTS: 1 REF: 4.3 STA: 2.A | 6.A KEY: polynomial function | quartic function | quintic function 11. ANS: g (x)  f (x  1)

TOP: Skills Practice

PTS: 1 REF: 4.3 STA: 2.A | 6.A KEY: polynomial function | quartic function | quintic function 12. ANS: g (x)  f (x  1)  3

TOP: Skills Practice

PTS: 1 REF: 4.3 STA: 2.A | 6.A KEY: polynomial function | quartic function | quintic function

TOP: Skills Practice

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