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

How do you graph rational functions step by step?

Factor the numerator and denominator, then note the x-values that make the original denominator zero. Cancel shared factors to find holes; the remaining denominator zeros give vertical asymptotes. Compare degrees to find a horizontal or slant asymptote, then find the intercepts. Finally, plot a few test points on each side of every vertical asymptote to see which way each branch curves.

What is a hole in graphing rational functions?

A hole is a single missing point on the graph, caused by a factor that cancels from both the numerator and the denominator. For g(x) = (x² − 9)/(x − 3), factoring gives (x − 3)(x + 3)/(x − 3), which simplifies to x + 3. The graph is the line y = x + 3 with an open circle at (3, 6), because x = 3 is still not allowed.

How do you find the horizontal asymptote when graphing rational functions?

Compare the degree of the numerator with the degree of the denominator. If the top degree is smaller, the horizontal asymptote is y = 0. If the degrees are equal, divide the leading coefficients: for (2x + 1)/(x − 4) the asymptote is y = 2. If the top degree is larger, there is no horizontal asymptote, though there may be a slant one.

How do you graph rational functions with slant asymptotes?

A slant (oblique) asymptote appears when the numerator’s degree is exactly one more than the denominator’s. Divide the polynomials and keep the quotient, ignoring the remainder. For f(x) = (x² + 1)/x, dividing gives x + 1/x, so the slant asymptote is y = x and the vertical asymptote is x = 0. Sketch both lines first, then let the branches approach them.

How do you graph rational functions on a TI-84 Plus CE?

Press Y=, type the function with parentheses around the whole numerator and the whole denominator, such as (X+1)/(X−2), then press GRAPH. Use the TABLE to check values near the asymptote at x = 2. Depending on settings, the calculator may draw a steep line across a vertical asymptote; that line is not part of the graph, so confirm asymptotes and holes algebraically.

Is graphing rational functions hard?

It feels hard at first because several features combine in one sketch, but it becomes routine once you follow the same order every time. Most mistakes come from canceling a factor and forgetting it leaves a hole, or from skipping test points near a vertical asymptote. Practicing with simple examples like (x + 1)/(x − 2) builds confidence before slant asymptotes.

Related Topics

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