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

$$\{x \mid x \neq 1, x \neq -1\}$$

Question 1

Which of the following is the correct vertex form of the quadratic function $$f(x) = x^2$ + 6x + 7$$?

Accepted Answer

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

Question 2

Given the polynomial $$f(x) = 2x^4$ - 12x^3$ + 18x^2$, what are the zeros and their multiplicities?

Accepted Answer

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

Question 3

A rational function is given by $$f(x) = \frac{x^2$ - 4}{x^2$ - x - 6}. $$What are the vertical asymptotes and any holes in the graph?

Accepted Answer

Vertical asymptote at $$x = 3$$; hole at $$x = -2$$

Question 4

Which of the following best describes the end behavior of $$f(x) = -2x^5$ + 3x^2$ - 1$$?

Accepted Answer

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

Question 5

What is the domain of the rational function $$f(x) = \frac{2x^2$ + 3x - 2}{x^2$ - 1}$$?

Accepted Answer

$$\{x \mid x 
eq 1, x 
eq -1\}$$

Question 6

Given $$f(x) = (x - 1)^2$(x + 2)$$, which statement about the graph at $$x = 1 is $$correct?

Accepted Answer

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

Question 7

What is the horizontal asymptote of $$f(x) = \frac{3x^2$ + 5}{2x^2$ - 7}$$?

Accepted Answer

$$y = \frac{3}{2}$$

Question 8

A company models its profit with $$P(x) = -x^2$ + 6x + 5. $$What is the maximum profit, and at what value of $$x$$ does it occur?

Accepted Answer

Maximum profit is $$14 at$$ $$x = 3$$

Question 9

Which of the following is NOT a polynomial function?

Accepted Answer

$$f(x) = 2x$$^{-1}$$ + 3$$

Question 10

For the rational function $$f(x) = \frac{x^2$ + 4x + 3}{x^2$ - 9}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 3$$ and $$x = -3$$

Question 11

What is the maximum number of turning points for the polynomial $$f(x) = 4x^5$ - 2x^3$ + x$$?

Accepted Answer

4

Question 12

Which transformation describes the graph of $$f(x) = -(x + 2)^2$ + 1$$ compared to $$f(x) = x^2$?

Accepted Answer

Shift left 2 units, reflect over x-axis, shift up 1 unit

Question 13

Given $$f(x) = \frac{x^2$ - 9}{x^2$ - 4}$$, what is the lowest terms expression and the domain?

Accepted Answer

$$f(x) = \frac{(x + 3)(x - 3)}{(x + 2)(x - 2)}$$, domain: $$\{x \mid x 
eq 2, x 
eq -2\}$$

Question 14

If $$f(x) = (x + 1)^3$(x - 2)^2$, what does the graph do at x = -1$ and x = 2$?

Accepted Answer

Crosses at $$x = -1$$, touches at $$x = 2$$

Question 15

Which of the following rational functions has a hole at $$x = 2$$?

Accepted Answer

$$f(x) = \frac{(x - 2)(x + 1)}{(x - 2)(x - 3)}$$

Question 16

A rational function $$f(x) = \frac{2x^3$ - 6x^2$ + 6x - 2}{x - 1} is $$given. What is the y-intercept?

Accepted Answer

$$f(0) = \frac{-2}{-1} = 2$$

Question 17

Which of the following best describes the range of $$f(x) = x^2$?

Accepted Answer

$$[0, \infty)$$

Question 18

Given $$f(x) = \frac{1}{x - 3} + 2$$, what are the vertical and horizontal asymptotes?

Accepted Answer

Vertical asymptote at $$x = 3$$, horizontal asymptote at $$y = 2$$

Question 19

Which of the following describes the transformation from $$f(x) = \frac{1}{x^2$} to$$ $$g(x) = \frac{1}{(x - 2)^2$} + 3$$?

Accepted Answer

Shift right 2 units and up 3 units

Polynomial and Rational Functions: Quadratic Functions, Properties, Graphing, and Asymptotes

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