Q: Given the polynomial function $$f(x) = 2x^3$ - 5x^2$ + 8x - 2$$, which of the following best describes its end behavior?

As $$x \to \infty$$, $$f(x) \to \infty$$; as $$x \to -\infty$$, $$f(x) \to -\infty$$

Q: A polynomial $$f(x)$$ has $$x-$$intercepts at $$x = -1, 2, 3$$ and passes through the point $$(0, 6). $$Which of the following could be the equation for $$f(x)$$?

$$f(x) = -2(x+1)(x-2)(x-3)$$, \text{ where } -2(x+1)(x-2)(x-3) = 6 \text{ when } x=0$

Question 1

Given the polynomial function $$f(x) = 2x^3$ - 5x^2$ + 8x - 2$$, which of the following best describes its end behavior?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o \infty$$; as $$x 	o -\infty$$, $$f(x) 	o -\infty$$

Question 2

Which of the following is the correct set of $$x-$$intercepts for the polynomial $$f(x) = x^3$ - 4x$$?

Accepted Answer

$$x = -2, 0, 2$$

Question 3

A polynomial function $$f(x)$$ has a zero at $$x = 2$$ with multiplicity 3. What does the graph do at $$x = 2$$?

Accepted Answer

It crosses the $$x-$$axis at $$x = 2$$

Question 4

Given $$f(x) = x^4$ - 2x^2$ + 1$$, what is the minimum value of $$f(x)$$?

Accepted Answer

$$1$$

Question 5

Which of the following intervals should be used to analyze the unknown behavior of the graph of $$f(x) if $$the known $$x-$$values are $$-3, 0, 2, 5$$?

Accepted Answer

$$(-\infty, -3), (-3, 0), (0, 2), (2, 5), (5, \infty)$$

Question 6

If $$f(x) = x^3$ - 6x^2$ + 11x - 6$$, which of the following is a turning point of the graph?

Accepted Answer

$$(1.5, f(1.5))$$

Question 7

Which of the following best describes the domain and range of any polynomial function?

Accepted Answer

Domain: $$(-\infty, \infty)$$, Range: depends on the degree and leading coefficient

Question 8

A polynomial $$f(x)$$ has $$x-$$intercepts at $$x = -1, 2, 3$$ and passes through the point $$(0, 6). $$Which of the following could be the equation for $$f(x)$$?

Accepted Answer

$$f(x) = -2(x+1)(x-2)(x-3)$$, 	ext{ where } -2(x+1)(x-2)(x-3) = 6 	ext{ when } x=0$

Question 9

Which of the following statements about the graph of a cubic polynomial is always true?

Accepted Answer

Its end behavior is opposite on each side.

Question 10

Given $$f(x) = x^4$ - 4x^2$ + 3$$, which of the following is a $$y-$$intercept of the graph?

Accepted Answer

$$(0, 3)$$

Question 11

If a polynomial function has a zero at $$x = 1$$ with even multiplicity, what does the graph do at $$x = 1$$?

Accepted Answer

It touches the $$x-$$axis and turns around at $$x = 1$$

Question 12

Which of the following best describes the process for graphing a polynomial function?

Accepted Answer

Find end behavior, $$x-$$intercepts, $$y-$$intercept, turning points, and plot points between.

Question 13

Given $$f(x) = x^3$ - 3x^2$ - 4x + 12$$, which of the following is a correct factorization?

Accepted Answer

$$f(x) = (x-2)(x+2)(x-3)$$

Question 14

A real-world scenario: The height $$h (in $$meters) of a projectile at time $$t (in $$seconds) is modeled by $$h(t) = -5t^2$ + 20t + 1. $$What is the maximum height reached by the projectile?

Accepted Answer

$$25$$ meters

Question 15

Which of the following is NOT a possible graph for a quartic polynomial?

Accepted Answer

A graph with only one $$x-$$intercept and end behavior in opposite directions

Graphing Polynomial Functions: Intervals, End Behavior, and Key Features

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