AP PreCalculus Practice Quiz: Rational Functions and Vertical Asymptotes

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 9 questions to check your progress.

Question 1 of 9

Which of the following is the vertical asymptote for the graph of the rational function r(x) = 1 / (x - 5)?

A)x = -5B)y = 5C)x = 5D)x = 1

All Questions (9)

Which of the following is the vertical asymptote for the graph of the rational function r(x) = 1 / (x - 5)?

A) x = -5

B) y = 5

C) x = 5

D) x = 1

Correct Answer: C

A vertical asymptote occurs where the denominator is zero and the numerator is not. The denominator, x - 5, is zero when x = 5. The numerator is 1 (a non-zero constant). Therefore, the vertical asymptote is at x = 5.

Consider the rational function f(x) = (x - 2) / (x^2 - 4). Which of the following statements correctly identifies the vertical asymptote(s) of the graph of f(x)?

A) The graph has vertical asymptotes at x = 2 and x = -2.

B) The graph has a vertical asymptote at x = -2 only.

C) The graph has a vertical asymptote at x = 2 only.

D) The graph has no vertical asymptotes.

Correct Answer: B

The denominator x^2 - 4 factors to (x - 2)(x + 2), so its real zeros are x = 2 and x = -2. The numerator's real zero is x = 2. Since x = 2 is a zero of both the numerator and the denominator, it corresponds to a hole, not a vertical asymptote. The value x = -2 is a zero of the denominator but not the numerator, so the only vertical asymptote is at x = -2.

The graph of a rational function r(x) has a vertical asymptote at x = a. According to the provided content, what behavior is expected for the values of r(x) as x gets closer to a?

A) The values of r(x) approach a single finite number.

B) The values of r(x) approach zero.

C) The values of r(x) increase or decrease without bound.

D) The values of r(x) approach a.

Correct Answer: C

The provided content states: 'Near a vertical asymptote, x = a, of a rational function, the values of the rational function r increase or decrease without bound.' This means the function's output goes towards positive or negative infinity.

A rational function is defined as r(x) = p(x) / q(x). If q(a) = 0 and p(a) ≠ 0 for some real number a, what feature must the graph of r(x) have at x = a?

A) An x-intercept

B) A vertical asymptote

C) A hole in the graph

D) A horizontal asymptote

Correct Answer: B

The provided rule states that if 'a' is a real zero of the denominator (q(a) = 0) and is not also a real zero of the numerator (p(a) ≠ 0), then the graph has a vertical asymptote at x = a. This directly matches the conditions given in the question.

What are the equations of all vertical asymptotes of the graph of the function g(x) = (x + 1) / (x^2 - 2x - 8)?

A) x = -2 and x = 4

B) x = 2 and x = -4

C) x = -1

D) x = 8

Correct Answer: A

To find the vertical asymptotes, we must find the zeros of the denominator. The denominator x^2 - 2x - 8 factors to (x - 4)(x + 2). The zeros are x = 4 and x = -2. The zero of the numerator is x = -1. Since neither of the denominator's zeros are also zeros of the numerator, the vertical asymptotes are at x = -2 and x = 4.

Which of the following rational functions has a graph with a vertical asymptote at x = 3?

A) r(x) = (x - 3) / (x + 3)

B) r(x) = (x + 3) / (x - 3)

C) r(x) = (x - 3) / (x^2 - 9)

D) r(x) = 1 / (x^2 + 9)

Correct Answer: B

A vertical asymptote occurs at x = 3 if the denominator is zero at x = 3 and the numerator is not. In option B, the denominator (x - 3) is zero at x = 3, and the numerator (x + 3) is 6, which is not zero. Option A has an asymptote at x=-3. Option C has a hole at x=3. Option D has no real zeros in the denominator.

The values of a rational function r(x) increase without bound as x approaches -1 from the left and from the right. What feature does the graph of r(x) have at x = -1?

A) A zero

B) A y-intercept

C) A horizontal asymptote

D) A vertical asymptote

Correct Answer: D

The provided content explains that near a vertical asymptote, the values of the rational function increase or decrease without bound. The behavior described in the question directly matches this definition for a vertical asymptote at x = -1.

A rational function is given by h(x) = (x^2 + 5x + 6) / (x^2 + 3x + 2). Identify the vertical asymptote(s) of the graph of h(x).

A) x = -1 and x = -2

B) x = -2 and x = -3

C) x = -1 only

D) x = -3 only

Correct Answer: C

First, factor the numerator and the denominator. The numerator is (x + 2)(x + 3), with zeros at x = -2 and x = -3. The denominator is (x + 2)(x + 1), with zeros at x = -2 and x = -1. A vertical asymptote occurs at a zero of the denominator that is not also a zero of the numerator. The value x = -1 is a zero of the denominator but not the numerator. The value x = -2 is a zero of both, which creates a hole, not an asymptote. Therefore, the only vertical asymptote is at x = -1.

According to the provided text, a vertical asymptote for a rational function is found by identifying the real zeros of which part of the function?

A) The polynomial function in the numerator.

B) The polynomial function in the denominator.

C) Both the numerator and the denominator.

D) The constant terms of the function.

Correct Answer: B

The provided text explicitly states that a vertical asymptote occurs at x = a if 'a is a real zero of the polynomial function in the denominator' (and not also a zero of the numerator). The primary condition is that the value must make the denominator zero.