Precalculus Rational Functions and Asymptotes Lesson (Free PDF)

Precalculus math lesson: Rational Functions and Asymptotes – Maths PDF

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This rational functions lesson for Precalculus covers rational functions and asymptotes (Common Core A-APR.D.7, F-IF.C.7d, A-CED.A.1), with a free PDF to print.

A rational function is a quotient of polynomials p(x)/q(x). Zeros of the denominator that do not cancel give vertical asymptotes, and factors that cancel leave holes. Comparing degrees gives the end behavior: a horizontal asymptote when the numerator’s degree is not larger, and a slant asymptote when it is exactly one larger.

Precalculus · Common Core State Standards: A-APR.D.7, F-IF.C.7d, A-CED.A.1

Key facts in this rational functions lesson

Key facts

Domain: all real numbers except zeros of the denominator.
Vertical asymptote: a denominator zero that does not cancel. Hole: a cancelled factor.
Degree of numerator < denominator: y = 0. Equal degrees: y = ratio of leading coefficients.
Degree of numerator = degree of denominator + 1: slant asymptote from division.
Always check rational equation answers for extraneous solutions.

What is a rational function?

A rational function is a quotient of two polynomials, \(f(x)=\frac{p(x)}{q(x)}\) with \(q(x)\ne 0\). Its domain excludes every zero of \(q\). Rational functions can have breaks: vertical asymptotes, holes, and horizontal or slant asymptotes that describe the end behavior.

How do you find asymptotes and holes?

Factor numerator and denominator completely.
A factor that cancels gives a hole at that \(x\)-value.
A zero of the denominator that does not cancel gives a vertical asymptote there.
Compare the degrees of the numerator (\(n\)) and denominator (\(m\)) to find the end behavior.

Degrees	End behavior asymptote
\(n< m\)	Horizontal asymptote \(y=0\)
\(n=m\)	Horizontal asymptote \(y=\) ratio of leading coefficients
\(n=m+1\)	Slant asymptote: the quotient (without the remainder) from division
\(n>m+1\)	No horizontal or slant asymptote

Example: For \(f(x)=\frac{2x}{x-3}\), the denominator is 0 at \(x=3\), so there is a vertical asymptote there. The degrees are equal and the leading coefficients are 2 and 1, so the horizontal asymptote is \(y=2\).

Dashed lines show the asymptotes \(x=3\) and \(y=2\).

For \(f(x)=\frac{x^{2}-4}{x-2}=\frac{(x-2)(x+2)}{x-2}\), the factor \(x-2\) cancels. The graph is the line \(y=x+2\) with a hole at \(x=2\), where \(y=4\).

A hole is a single missing point, drawn as an open circle.

How do you find a slant asymptote?

When the numerator’s degree is one more than the denominator’s, divide. For \(\frac{x^{2}-x-2}{x-3}\), the quotient is \(x+2\) with remainder 4, so \(f(x)=x+2+\frac{4}{x-3}\). As \(|x|\) grows, the fraction goes to 0, and the graph approaches the line \(y=x+2\).

A rational function can approach a slanted line.

How do you solve rational equations?

Multiply both sides by the least common denominator to clear fractions, solve, then check every answer in the original equation. A solution that makes a denominator 0 is extraneous and must be rejected. For rational inequalities, move everything to one side, factor, and use a sign chart with the zeros of the numerator and denominator.

Solution of \(\frac{x-1}{x+2}\ge 0\): the zero \(1\) is included (closed) and the denominator zero \(-2\) is excluded (open).

Intercepts: the \(x\)-intercepts are zeros of the numerator (that do not cancel), and the \(y\)-intercept is \(f(0)\).

Rational functions lesson: questions and answers

What is the difference between a hole and a vertical asymptote?

Both come from a zero of the denominator. If the factor cancels with the numerator, the graph has a hole (a single missing point). If it does not cancel, the graph has a vertical asymptote.

How do I find a horizontal asymptote?

Compare degrees. If the numerator has the lower degree, the asymptote is y = 0. If the degrees are equal, divide the leading coefficients. If the numerator has the higher degree, there is no horizontal asymptote.

Can a graph cross a horizontal asymptote?

Yes, in the middle of the graph. A horizontal asymptote describes the behavior as x gets very large or very small, not the behavior at every x.

What is an extraneous solution?

It is a value that appears when you solve but makes a denominator zero in the original equation. Always substitute your answers back and reject any that fail.

How do I find intercepts of a rational function?

Set the numerator equal to 0 for x-intercepts, ignoring values that also make the denominator 0. For the y-intercept, compute f(0).

How to use this rational functions lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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