College Algebra

Rational Functions and Asymptotes

Lesson

A rational function is a quotient of two polynomials, $f(x) = \dfrac{p(x)}{q(x)}$ . The graph has two kinds of asymptotes — invisible lines the graph approaches but never crosses (at least, not at the ends).

Vertical asymptotes (VAs): the graph blows up to $\pm\infty$ at certain $x$ values. They occur where the denominatoris zero (and the numerator isn’t zero at the same spot).

Factor the denominator.
Set each factor equal to zero.
Each solution is a vertical asymptote.

Horizontal asymptotes (HAs): what the graph approaches as $x \to \pm\infty$ . Decided by comparing the degrees of numerator and denominator:

Numerator degree < denominator degree: horizontal asymptote at $y = 0$ .
Numerator degree = denominator degree: horizontal asymptote at $y =$ the ratio of leading coefficients.
Numerator degree > denominator degree: no horizontal asymptote.

Worked example 1 — VAs

f(x) = \frac{x + 1}{x^2 - 9}

Factor the denominator: $x^2 - 9 = (x - 3)(x + 3)$ . Set each factor to zero:

x = 3 \quad\text{or}\quad x = -3

Vertical asymptotes at $x = 3$ and $x = -3$ .

Worked example 2 — HA

f(x) = \frac{2x - 5}{3x + 1}

Numerator and denominator both have degree 1. Take the ratio of leading coefficients:

y = \frac{2}{3}

How to type your answer

Each problem asks for a vertical asymptote ( $x = ?$ ) or a horizontal asymptote ( $y = ?$ ). Type the value(s), comma-separated if there’s more than one. Examples: 3, 2,-3, 0, 2/3.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

f(x) = \dfrac{1}{x - 3};\ \text{vertical asymptote: } x = ?

Problem 2

f(x) = \dfrac{1}{x};\ \text{horizontal asymptote: } y = ?

Problem 3

f(x) = \dfrac{2}{x + 5};\ \text{vertical asymptote: } x = ?

Problem 4

f(x) = \dfrac{3x + 1}{x};\ \text{horizontal asymptote: } y = ?

Practice

Standard problems matching the lesson.

Problem 5

f(x) = \dfrac{1}{(x - 2)(x + 4)};\ \text{vertical asymptotes: } x = ?

Problem 6

f(x) = \dfrac{2x - 5}{3x + 1};\ \text{horizontal asymptote: } y = ?

Problem 7

f(x) = \dfrac{x + 1}{x^2 - 9};\ \text{vertical asymptotes: } x = ?

Problem 8

f(x) = \dfrac{x + 1}{x^2 - 9};\ \text{horizontal asymptote: } y = ?

Problem 9

f(x) = \dfrac{3}{x^2 + 5x + 6};\ \text{vertical asymptotes: } x = ?

Problem 10

f(x) = \dfrac{5x^2}{x^2 + 3};\ \text{horizontal asymptote: } y = ?

Problem 11

f(x) = \dfrac{x + 5}{x^2 - 4};\ \text{vertical asymptotes: } x = ?

Problem 12

f(x) = \dfrac{4x + 7}{2x - 1};\ \text{horizontal asymptote: } y = ?

Problem 13

f(x) = \dfrac{1}{x^2 - x - 6};\ \text{vertical asymptotes: } x = ?

Problem 14

f(x) = \dfrac{x^2 - 4}{2x^2 + 1};\ \text{horizontal asymptote: } y = ?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

f(x) = \dfrac{1}{2x - 6};\ \text{vertical asymptote: } x = ?

Problem 16

f(x) = \dfrac{3x^2 + x}{6x^2 - 2};\ \text{horizontal asymptote: } y = ?

Problem 17

f(x) = \dfrac{5}{x^2 - 7x + 10};\ \text{vertical asymptotes: } x = ?

Problem 18

f(x) = \dfrac{-2x + 1}{5x + 3};\ \text{horizontal asymptote: } y = ?

Problem 19

f(x) = \dfrac{1}{x^3 - x};\ \text{vertical asymptotes: } x = ?

Problem 20

f(x) = \dfrac{7x^2 - x}{2x^2 + x + 1};\ \text{horizontal asymptote: } y = ?

Problem 21

f(x) = \dfrac{2}{x^3 - 4x};\ \text{vertical asymptotes: } x = ?

Problem 22

f(x) = \dfrac{3x + 5}{x^2 - 1};\ \text{horizontal asymptote: } y = ?