Q: Given the polynomial $$f(x) = -2x^5$ + 3x^3$ - x + 7$$, what is the end behavior of the graph?

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

Q: Given $$f(x) = -x^4$ + 2x^2$ - 5$$, what is the end behavior of the graph?

As $$x \to \pm\infty$$, $$f(x) \to -\infty$$

Question 1

Which of the following is the correct definition of a polynomial function of degree $$n$$?

Accepted Answer

$$f(x) = a_n x^n + a_{n-1} x$$^{n-1}$$ + \ldots + a_1$ x + a_0$, where a_n$$ 
eq 0$$ and all coefficients are real numbers.

Question 2

Which of the following statements about the graph of a polynomial function is TRUE?

Accepted Answer

The graph of a polynomial function is always smooth and continuous, with no sharp corners or breaks.

Question 3

Given the polynomial $$f(x) = -2x^5$ + 3x^3$ - x + 7$$, what is the end behavior of the graph?

Accepted Answer

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

Question 4

Which of the following polynomials has a graph that rises to the left and right as $$x$$ approaches $$\pm\infty$$?

Accepted Answer

$$f(x) = 2x^4$ - 5x^2$ + 1$$

Question 5

Suppose $$f(x) = (x-1)^2$(x+3). $$Which statement about the zeros and their multiplicities is correct?

Accepted Answer

$$x = 1 is a $$zero of multiplicity 2 (the graph touches and turns at $$x=1$$), $$x = -3 is a $$zero of multiplicity 1 (the graph crosses at $$x=-3).$$

Question 6

A chemical reaction's rate can be modeled by $$f(x) = x^3$ - 6x^2$ + 9x. $$Find all real zeros and state their multiplicities.

Accepted Answer

$$x = 0 ($$multiplicity 1), $$x = 3 ($$multiplicity 2)

Question 7

If $$f(x) = x^4$ - 16$$, which of the following correctly lists all real zeros and their multiplicities?

Accepted Answer

Question 8

A company models its profit with $$f(x) = x^3$ - 4x. $$Use the Intermediate Value Theorem to show that there is a real zero between $$x = 1$$ and $$x = 2.$$

Accepted Answer

$$f(1) = -3$$, $$f(2) = 4$$, since $$f(1)  0$$, there is a zero between $$x=1$$ and $$x=2.$$

Question 9

How many turning points can the graph of a degree 6 polynomial have at most?

Accepted Answer

5

Question 10

Which of the following functions is NOT a polynomial function?

Accepted Answer

$$f(x) = x$$^{-2}$$ + 3x$$

Question 11

A polynomial $$f(x)$$ has a zero at $$x = 4$$ with multiplicity 3. What does the graph do at $$x = 4$$?

Accepted Answer

It crosses the $$x-$$axis and flattens out at $$x = 4.$$

Question 12

Given $$f(x) = -x^4$ + 2x^2$ - 5$$, what is the end behavior of the graph?

Accepted Answer

As $$x 	o \pm\infty$$, $$f(x) 	o -\infty$$

Question 13

Which of the following describes the symmetry of $$f(x) = x^4$ - 2x^2$ + 1$$?

Accepted Answer

The function is even; it has $$y-$$axis symmetry.

Question 14

A population model is given by $$f(x) = (x-2)^2$(x+1). $$Which statement is true about the behavior at $$x = -1$$?

Accepted Answer

The graph crosses the $$x-$$axis at $$x = -1.$$

Question 15

If $$f(x) = x^3$ - 3x^2$ + 2x$$, what is the $$y-$$intercept of the graph?

Accepted Answer

$$0$$

Question 16

Which of the following is a correct application of the Leading Coefficient Test for $$f(x) = -x^3$ + 2x^2$ - 5$$?

Accepted Answer

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

Question 17

A polynomial function $$f(x)$$ has degree 5. What is the maximum number of $$x-$$intercepts it can have?

Accepted Answer

5

Question 18

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

Accepted Answer

A graph with a hole at $$x = 2$$

Polynomial Functions and Their Graphs: Definitions, Properties, and Graphing Strategies

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