How to Graph Rational Functions?

Graphing rational functions requires a systematic approach: find the asymptotes (the lines the graph approaches but never crosses), locate any x- and y-intercepts, identify holes, and then plot a few key points to see which way the branches curve. Once you master the process, graphing rational functions becomes a reliable, step-by-step procedure.

What Is a Rational Function?

A rational function is a ratio of two polynomials: \(\color{blue}{f(x) = \frac{P(x)}{Q(x)}}\), where Q(x) ≠ 0. Common examples include \(\color{blue}{f(x) = \frac{1}{x}}\), \(\color{blue}{f(x) = \frac{2}{(x – 3)}}\), and \(\color{blue}{f(x) = \frac{(x + 1)}{(x – 2)}}\). The graph of a rational function often has gaps, breaks, or asymptote lines that distinguish it from polynomial graphs.

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How to Graph a Rational Function

Step 1 — Find the vertical asymptote(s)

Set the denominator \(\color{blue}{Q(x) = 0}\) and solve. Each solution gives a vertical asymptote (a vertical line the graph never crosses).

Example: \(\color{blue}{f(x) = \frac{2}{(x – 3)}}\). \(\color{blue}{\text{ Denominator } = 0}\) when \(\color{blue}{x = 3}\). Vertical asymptote: \(\color{blue}{x = 3}\).

Step 2 — Find the horizontal asymptote

Compare the degrees of the numerator (n) and denominator (m):

• n < m: horizontal asymptote \(\color{blue}{y = 0}\).
• \(\color{blue}{n = m}\): horizontal asymptote \(\color{blue}{y = \frac{(\text{ leading coefficient of P })}{(\text{ leading coefficient of Q })}}\).
• n > m: no horizontal asymptote (there may be a slant asymptote instead).

Example: \(\color{blue}{f(x) = \frac{2}{(x – 3)}}\). Degree of \(\color{blue}{\text{ numerator } = 0}\), degree of \(\color{blue}{\text{ denominator } = 1}\). n < m, so horizontal asymptote: \(\color{blue}{y = 0}\).

Step 3 — Find x-intercepts and y-intercept

x-intercepts: Set \(\color{blue}{P(x) = 0}\).
y-intercept: Substitute \(\color{blue}{x = 0}\) into f(x).

Example: \(\color{blue}{f(x) = \frac{(x + 1)}{(x – 2)}}\).
x-intercept: \(\color{blue}{x + 1 = 0}\) → x = −1. Point (−1, 0).
y-intercept: \(\color{blue}{f(0) = \frac{1}{(-2)}}\) = −\(\color{blue}{\frac{1}{2}}\). Point (0, −\(\color{blue}{\frac{1}{2}}\)).

Step 4 — Check for holes

Factor the numerator and denominator. If a factor cancels, the cancelled factor creates a hole at the x-value that makes it zero.

Step 5 — Plot additional points and sketch

Choose test x-values on each side of every vertical asymptote to determine the direction of the branches, then sketch the curve approaching the asymptotes.

Step-by-Step Summary

1. Factor numerator and denominator; cancel common factors (note holes).
2. Find vertical asymptotes (\(\color{blue}{\text{ denominator } = 0}\)).
3. Find horizontal (or slant) asymptote (compare degrees).
4. Find x-intercepts (\(\color{blue}{\text{ numerator } = 0}\)) and y-intercept (\(\color{blue}{x = 0}\)).
5. Plot test points; sketch the graph approaching the asymptotes.

Watch: Graphing Rational Functions with Asymptotes (Video Lesson)

The Organic Chemistry Tutor provides a comprehensive guide to graphing rational functions, including vertical, horizontal, and slant asymptotes, holes, and domain and range:

Graphing Rational Functions – Worked Examples

Example 1: Graph \(\color{blue}{f(x) = \frac{2}{(x – 3)}}\).

Vertical asymptote: \(\color{blue}{x = 3}\).
Horizontal asymptote: \(\color{blue}{y = 0}\) (degree 0 < degree 1).
x-intercept: none (\(\color{blue}{\text{ numerator } = 2}\) never equals 0).
y-intercept: \(\color{blue}{f(0) = \frac{2}{(-3)}}\) = −\(\color{blue}{\frac{2}{3}}\) ≈ −0.67.
Test \(\color{blue}{x = 4}\): \(\color{blue}{f(4) = \frac{2}{1} = 2}\) (above x-axis to right of VA); \(\color{blue}{x = 2}\): \(\color{blue}{f(2) = \frac{2}{(-1)}}\) = −2 (below x-axis to left of VA).

Example 2: Graph \(\color{blue}{f(x) = \frac{(x + 1)}{(x – 2)}}\).

Vertical asymptote: \(\color{blue}{x = 2}\).
Horizontal asymptote: \(\color{blue}{y = 1}\) (both degree 1; leading coefficients \(\color{blue}{\frac{1}{1}}\)).
x-intercept: x = −1. Point (−1, 0).
y-intercept: \(\color{blue}{f(0) = \frac{1}{(-2)}}\) = −\(\color{blue}{\frac{1}{2}}\). Point (0, −\(\color{blue}{\frac{1}{2}}\)).

Example 3: Find the hole in \(\color{blue}{f(x) = \frac{(x^{2} – 4)}{(x + 2)}}\).

Factor: \(\color{blue}{\frac{(x + 2)(x – 2)}{(x + 2)}}\). Cancel (\(\color{blue}{x + 2}\)): simplified form is \(\color{blue}{x – 2}\), with a hole at x = −2. The hole is at (−2, −4).

Example 4: Find all asymptotes of \(\color{blue}{f(x) = \frac{3x}{(x^{2} – 9)}}\).

Factor denominator: (\(\color{blue}{x – 3}\))(\(\color{blue}{x + 3}\)). Vertical asymptotes: \(\color{blue}{x = 3}\), x = −3.
Degrees: numerator degree 1 < denominator degree 2. Horizontal asymptote: \(\color{blue}{y = 0}\).

More Practice: Step-by-Step Graphing Guide (Video Lesson)

Mario’s Math Tutoring walks through three complete examples of graphing rational functions from start to finish:

Exercises for Graphing Rational Functions

For each function, find (a) vertical asymptotes, (b) horizontal asymptote, (c) x-intercepts, (d) y-intercept.

1. \(\color{blue}{f(x) = \frac{1}{(x + 4)}}\)
2. \(\color{blue}{f(x) = \frac{(x – 3)}{(x + 1)}}\)
3. \(\color{blue}{f(x) = \frac{2x}{(x^{2} – 1)}}\)
4. \(\color{blue}{f(x) = \frac{(x^{2} – 9)}{(x – 3)}}\)
5. \(\color{blue}{f(x) = \frac{4}{(x^{2} + 4)}}\)

Answers

1. (a) x = −4; (b) \(\color{blue}{y = 0}\); (c) none; (d) \(\color{blue}{f(0) = \frac{1}{4}}\)
2. (a) x = −1; (b) \(\color{blue}{y = 1}\); (c) \(\color{blue}{x = 3}\); (d) f(0) = −3
3. (a) \(\color{blue}{x = 1}\), x = −1; (b) \(\color{blue}{y = 0}\); (c) \(\color{blue}{x = 0}\); (d) \(\color{blue}{f(0) = 0}\)
4. Factor: \(\color{blue}{\frac{(x + 3)(x – 3)}{(x – 3)} = x + 3}\) (hole at \(\color{blue}{x = 3}\)); no VA; no HA (line); y-int \(\color{blue}{f(0) = 3}\)
5. (a) none (denominator \(\color{blue}{x^{2} + 4}\) > 0 always); (b) \(\color{blue}{y = 0}\); (c) none; (d) \(\color{blue}{f(0) = 1}\)

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Want More Practice?

We haven’t published a worksheet built specifically for Graphing Rational Functions just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work — they’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Graphing Functions and Transformations Worksheet
• Download Inverse Variation Worksheet
• Download Domain and Range Worksheet

Frequently Asked Questions

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

A hole (removable discontinuity) occurs when a factor cancels from both numerator and denominator; the graph is simply missing a single point. A vertical asymptote occurs when the denominator is zero but the factor does not cancel; the graph shoots toward ±∞ near that x-value.

Can a graph cross a horizontal asymptote?

Yes, for horizontal asymptotes. The graph may cross \(\color{blue}{y = k}\) somewhere in the middle; it just approaches \(\color{blue}{y = k}\) as x → ±∞. Vertical asymptotes can never be crossed.

How do I find a slant (oblique) asymptote?

A slant asymptote exists when the degree of the numerator is exactly 1 more than the degree of the denominator. Perform polynomial long division; the quotient (ignoring the remainder) is the equation of the slant asymptote.

Related Topics

• Simplifying Complex Fractions
• Adding and Subtracting Rational Expressions
• Multiplying Rational Expressions
• Dividing Rational Expressions
• Finding Domain and Range of a Function