Rational functions: vertical and horizontal asymptotes

Warm-up

Three quick questions from earlier lessons.

Discover

Move the sliders and watch the graph of $y = \dfrac{a}{x - h}$ .

The graph of y = a over (x − h): two branches separated by a vertical asymptote at x = h, approaching the x-axis far left and far right.

Where is the vertical line that the graph never touches?

At $x = h$
At $x = a$

Key idea

A rational function is a quotient of polynomials.

Vertical asymptotes are where the denominator is 0 (and the numerator is not). Horizontal asymptote: if the top degree is smaller,

y = 0

; if the degrees are equal,

y = \dfrac{\text{leading coefficient on top}}{\text{leading coefficient below}}

; if the top degree is larger, there is none.

Worked example

Find the asymptotes of $f(x) = \dfrac{6x + 1}{2x - 8}$ .

Vertical asymptote:

x = 4

Denominator 0: 2x − 8 = 0.

Horizontal asymptote:

y = \dfrac{6}{2} = 3

Same degree: ratio of leading coefficients.

Your turn, guided

Your turn: $f(x) = \dfrac{5x - 2}{x + 3}$ .

Vertical asymptote:

x = -3

x + 3 = 0.

Horizontal asymptote:

y = 5

Same degree: 5 over 1.

Practice ★

$f(x) = \dfrac{ 4x + 6 }{x - 6}$ has a vertical asymptote $x = c$ . What is $c$ ?
$f(x) = \dfrac{ 6x + 4 }{x - 5}$ has a vertical asymptote $x = c$ . What is $c$ ?
$f(x) = \dfrac{ 2x + 5 }{x - 2}$ has a vertical asymptote $x = c$ . What is $c$ ?
$f(x) = \dfrac{ 5x + 7 }{x - 4}$ has a vertical asymptote $x = c$ . What is $c$ ?

Watch out!

Practice ★★

$f(x) = \dfrac{ 5x + 6 }{ 2x + 4 }$ has a horizontal asymptote $y = k$ . What is $k$ ?
$f(x) = \dfrac{ 2x + 2 }{ 4x + 7 }$ has a horizontal asymptote $y = k$ . What is $k$ ?
$f(x) = \dfrac{ 8x + 5 }{ 4x + 5 }$ has a horizontal asymptote $y = k$ . What is $k$ ?

What is the horizontal asymptote of $f(x) = \dfrac{ 4x + 1}{x^2 + 6}$ ?
1. $y = 0$
2. $y = 4$
3. There is none, because the degrees are different
What is the horizontal asymptote of $f(x) = \dfrac{ 8x + 1}{x^2 + 5}$ ?
1. $y = 0$
2. $y = 8$
3. There is none, because the degrees are different

Exit ticket

Three questions to check you have got it.

$f(x) = \dfrac{ 5x + 6 }{ 2x + 4 }$ has a horizontal asymptote $y = k$ . What is $k$ ?
$f(x) = \dfrac{ 2x + 2 }{ 4x + 7 }$ has a horizontal asymptote $y = k$ . What is $k$ ?
$f(x) = \dfrac{ 8x + 5 }{ 4x + 5 }$ has a horizontal asymptote $y = k$ . What is $k$ ?

Summary card

Asymptotes of rational functions

Vertical: zeros of the denominator.
Horizontal: compare degrees. Top smaller: $y = 0$ . Equal: ratio of leading coefficients. Top larger: none.

Watch out: use the leading coefficients, not the constants.

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