Q: Given the polynomial function $$f(x) = -2x^4$ + 5x^3$ - x + 7$$, what is the end behavior of the function as $$x \to \infty$$ and $$x \to -\infty$$?

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

Q: If $$f(x) is a $$polynomial of degree 6 with leading coefficient negative, what is the end behavior?

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

Question 1

Given the polynomial function $$f(x) = -2x^4$ + 5x^3$ - x + 7$$, what is the end behavior of the function as $$x 	o \infty$$ and $$x 	o -\infty$$?

Accepted Answer

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

Question 2

Which of the following polynomial functions has a graph that rises to the right and falls to the left?

Accepted Answer

$$f(x) = x^3$ - 4x + 2$$

Question 3

The zeros of $$f(x) = (x-2)^3$(x+1)^2$ are:

Accepted Answer

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

Question 4

For the function $$f(x) = (x-3)^2$(x+2)$$, what happens at $$x = 3 on $$the graph?

Accepted Answer

The graph touches the x-axis at $$x = 3$$

Question 5

A polynomial function $$f(x) of $$degree 5 can have at most how many turning points?

Accepted Answer

$$4$$

Question 6

Given $$f(x) = x^3$ - 6x^2$ + 11x - 6$$, which of the following is NOT a zero of $$f(x)$$?

Accepted Answer

$$x = 6$$

Question 7

Which of the following statements about the graph of $$f(x) = (x+2)^4$(x-1) is $$true?

Accepted Answer

The graph touches the x-axis at $$x = -2$$ and crosses at $$x = 1$$

Question 8

If $$f(x) is a $$polynomial of degree 6 with leading coefficient negative, what is the end behavior?

Accepted Answer

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

Question 9

A chemical reaction's rate is modeled by $$f(x) = -x^3$ + 3x^2$ + 4x - 12. $$According to the Intermediate Value Theorem, if $$f(-2) = -38$$ and $$f(0) = -12$$, what can you conclude?

Accepted Answer

There is no zero between $$x = -2$$ and $$x = 0$$

Question 10

Which of the following functions is an even function?

Accepted Answer

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

Question 11

Suppose $$f(x) = (x-1)^2$(x+2)^3. $$Which statement is true about the graph at $$x = -2$?

Accepted Answer

The graph crosses the x-axis at $$x = -2$$

Question 12

A population model is given by $$P(t) = t^3$ - 6t^2$ + 9t. $$What is the y-intercept of the graph?

Accepted Answer

$$0$$

Question 13

Which of the following best describes the number of turning points for a degree $$n$$ polynomial?

Accepted Answer

At most $$n-1$$ turning points

Question 14

Given $$f(x) = x^4$ - 4x^2$, which of the following is NOT a zero of f(x$$)$$?

Accepted Answer

$$x = 4$$

Question 15

If $$f(x) = (x-1)^2$(x+3)$$, what is the degree of the polynomial?

Accepted Answer

$$3$$

Question 16

A polynomial $$f(x)$$ has zeros at $$x = -2 ($$multiplicity 2), $$x = 1 ($$multiplicity 1), and $$x = 3 ($$multiplicity 3). Which statement is true?

Accepted Answer

The graph touches the x-axis at $$x = -2$$ and crosses at $$x = 1$$ and $$x = 3$$

Question 17

Which of the following is a possible graph for a polynomial with degree 4 and a positive leading coefficient?

Accepted Answer

A graph that rises on both ends

Question 18

If $$f(x) = x^3$ - 2x^2$ - x + 2$$, use the Intermediate Value Theorem to determine if there is a zero between $$x = 0$$ and $$x = 1.$$

Accepted Answer

No, because $$f(0)$$ and $$f(1)$$ have the same sign

Question 19

Which of the following is true for the function $$f(x) = -x^5$ + 2x^3$ - x$$?

Accepted Answer

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

Study Guide: Polynomial Functions – End Behavior, Zeros, Multiplicity, and Graphing

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