Rational functions

Adapted from OpenStax College Algebra 2e¹

Asymptotes

The curve of $y = a + \frac{b}{c x + d}$ has a vertical asymptote $x = - \frac{c}{d}$ and a horizontal asymptote $y = a .$

Rectangular hyperbolas

Example

The curve of $y = 3 + \frac{1}{x + 2}$ has a vertical asymptote $x = - 2$ and a horizontal asymptote $y = 3 .$

We can then use a graphing calculator to sketch the curve, using our asymptotes to better understand the shape of the curve.

rectangular hyperbola 3+1/(x+2)

Curves of this form are rectangular hyperbolas, and appears frequently in our study.

Improper rational functions

A rational function is improper if the degree of the numerator is greater or equal to the degree of the denominator.

For example, $y = \frac{3 x + 7}{x + 2}$ is improper because the degree of both the numerator and denominator is $1 .$

Long division

To find the asymptotes of an improper rational function, we can use long division.

$\frac{3 x + 7}{x + 2}$ can be rewritten as $3 + \frac{1}{x + 2}$ via long division. We can now use this form to deduce that the asymptotes are $x = - 2$ and $y = 3 .$

Content in this page is adapted from OpenStax College Algebra 2e by Jay Abramson under the Creative Commons Attribution License.
Access for free at https://openstax.org/books/college-algebra-2e/pages/5-6-rational-functions ↩︎