Question 1

Which of the following best describes a vertical asymptote for a rational function?

Accepted Answer

A line $$x = a$$ where the function increases or decreases without bound as $$x$$ approaches $$a.$$

Question 2

Given the graph below ($$image_1$$), what are the equations of the horizontal and vertical asymptotes?

Accepted Answer

Horizontal asymptote: $$y = 0$$, Vertical asymptote: $$x = 0$$

Question 3

Consider the transformation $$y = \frac{1}{x-3} + 4. $$What are the equations of the vertical and horizontal asymptotes?

Accepted Answer

Vertical asymptote: $$x = 3$$, Horizontal asymptote: $$y = 4$$

Question 4

For the function $$f(x) = \frac{x^2$ - 4}{x^2$ + 3x - 10}$$, which of the following is a removable discontinuity?

Accepted Answer

$$x = 2$$

Question 5

Given $$f(x) = \frac{2}{5x + 2}$$, what is the domain of $$f(x)$$?

Accepted Answer

All real numbers except $$x = -\frac{2}{5}$$

Question 6

For $$f(x) = \frac{x^2$ + 2x - 3}{x^2$ + 5x - 6}$$, what are the vertical asymptotes?

Accepted Answer

$$x = -2$$ and $$x = -3$$

Question 7

Find the horizontal asymptote of $$f(x) = \frac{x^2$ + 2x - 3}{x^2$ + 5x - 6}.$$

Accepted Answer

$$y = 1$$

Question 8

What is the $$y-$$intercept of $$y = \frac{x^2$ + x - 6}{x^2$ + 3x - 4}$$?

Accepted Answer

$$(0, -6)$$

Question 9

Which of the following best describes a removable discontinuity in a rational function?

Accepted Answer

A point where the function is undefined due to a factor that cancels in the numerator and denominator.

Question 10

A rational function models the concentration of a drug in the bloodstream over time, $$C(t) = \frac{50}{t+2}. $$What is the horizontal asymptote and what does it represent in this context?

Accepted Answer

Horizontal asymptote: $$y = 0$$, representing the drug concentration approaches zero as time increases.

Question 11

If $$f(x) = \frac{2}{5x + 2}$$, what is the equation of the vertical asymptote?

Accepted Answer

$$x = -\frac{2}{5}$$

Question 12

For $$f(x) = \frac{x^2$ - 4}{x^2$ + 3x - 10}$$, what is the equation of the horizontal asymptote?

Accepted Answer

$$y = 1$$

Question 13

Given $$y = \frac{x^2$ + x - 6}{x^2$ + 3x - 4}$$, what are the $$x-$$intercepts?

Accepted Answer

$$x = 2$$ and $$x = -3$$

Question 14

A rational function $$f(x)$$ has a vertical asymptote at $$x = 5. $$Which of the following could be the denominator of $$f(x)$$?

Accepted Answer

$$x - 5$$

Question 15

Which transformation shifts the graph of $$y = \frac{1}{x}$$ three units right and four units up?

Accepted Answer

$$y = \frac{1}{x-3} + 4$$

Question 16

If a rational function has a higher degree in the denominator than in the numerator, what is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Precalculus Study Guide: Asymptotes and Rational Functions

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