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

$$\{ x \,|\, x \neq 3,\ x \neq -2 \}$$

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

$$\{ x \,|\, x \neq 2\sqrt{2},\ x \neq -2\sqrt{2} \}$$

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 quadratic function $$f(x) = -2(x + 1)^2$ + 5$$, what are the vertex and the axis of symmetry?

Accepted Answer

Vertex: $$(-1, 5)$$; Axis of symmetry: $$x = -1$$

Question 3

Which of the following functions is NOT a polynomial function?

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

Find all zeros and their multiplicities for $$f(x) = x^2$(x - 2)^3$ and state whether the graph crosses or touches the x-axis at each zero.

Accepted Answer

$$x = 0 ($$multiplicity 2, touches); $$x = 2 ($$multiplicity 3, crosses)

Question 6

What is the maximum number of turning points for the polynomial $$f(x) = 5x^6$ - 2x^3$ + x$$?

Accepted Answer

5

Question 7

A rational function is given by $$f(x) = \dfrac{x^2$ - 9}{x^2$ - x - 6}. $$What is the domain of $$f(x)$$?

Accepted Answer

$$\{ x \,|\, x 
eq 3,\ x 
eq -2 \}$$

Question 8

Which of the following describes the horizontal asymptote of $$f(x) = \dfrac{2x^2$ + 3}{4x^2$ - 5x + 1}$$?

Accepted Answer

$$y = \dfrac{2}{4}$$

Question 9

Find all vertical asymptotes and holes for $$f(x) = \dfrac{x^2$ - 4}{x^2$ - x - 6}.$$

Accepted Answer

Vertical asymptotes at $$x = 3$$ and $$x = -2$$; no holes

Question 10

Given $$f(x) = \dfrac{x + 2}{(x + 2)(x - 3)}$$, identify all vertical asymptotes and holes.

Accepted Answer

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

Question 11

Which transformation describes the graph of $$g(x) = \dfrac{1}{(x - 2)^2$} + 3$$ compared to $$f(x) = \dfrac{1}{x^2$}$$?

Accepted Answer

Shift right 2 units and up 3 units

Question 12

For the rational function $$f(x) = \dfrac{2x - 3}{x - 1}$$, what is the $$y-$$intercept?

Accepted Answer

$$(0, 3)$$

Question 13

A rational function $$f(x) = \dfrac{3x + 6}{x^2$ - 4} is $$given. What is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 14

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

Accepted Answer

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

Question 15

Suppose a rational function has a numerator and denominator of equal degree. What determines the horizontal asymptote?

Accepted Answer

The ratio of the leading coefficients

Question 16

Which of the following is the correct domain for $$f(x) = \dfrac{6x}{x^2$ - 8}$$?

Accepted Answer

$$\{ x \,|\, x 
eq 2\sqrt{2},\ x 
eq -2\sqrt{2} \}$$

Question 17

If $$f(x) = (3x + 2)(x - 1)^2$, what is the multiplicity of the zero at x = 1$ and does the graph cross or touch the x-axis there?

Accepted Answer

Multiplicity 2, touches the x-axis

Question 18

A rational function $$f(x) = \dfrac{x^2$ + 4x + 3}{x^2$ - 9} is $$given. What are the vertical asymptotes?

Accepted Answer

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

Question 19

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

Accepted Answer

$$[0, \infty)$$

Polynomial and Rational Functions: Quadratic Functions, Polynomial Graphs, and Rational Function Analysis

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