How to Graph Rational Expressions? (+FREE Worksheet!)

In this post, you will learn how to graph Rational Expressions. You can graph Rational Expressions in a few simple steps.

Related Topics

• How to Add and Subtract Rational Expressions
• How to Multiply Rational Expressions
• How to Divide Rational Expressions
• How to Solve Rational Equations
• How to Simplify Complex Fractions

A step-by-step guide to Graphing Rational Expressions

• A rational expression is a fraction in which the numerator and/or the denominator are polynomials. Examples: $\frac{1}{x},\frac{x^2}{x-1},\frac{x^2-x+2}{x^2+5x+1},\frac{m^2+6m-5}{m-2m}$
• To graph a rational function:

• Find the vertical asymptotes of the function if there are any. (Vertical asymptotes are vertical lines that correspond to the zeroes of the denominator. The graph will have a vertical asymptote at $x=a$ if the denominator is zero at $x=a$ and the numerator isn’t zero at $x=a$)
• Find the horizontal or slant asymptote. (If the numerator has a bigger degree than the denominator, there will be a slant asymptote. To find the slant asymptote, divide the numerator by the denominator using either long division or synthetic division.)
• If the denominator has a bigger degree than the numerator, the horizontal asymptote is the $x$-axes or the line $y=0$. If they have the same degree, the horizontal asymptote equals the leading coefficient (the coefficient of the largest exponent) of the numerator divided by the leading coefficient of the denominator.
• Find intercepts and plug in some values of $x$ and solve for $y$, then graph the function.

Examples

Graphing Rational Expressions – Example 1:

Graph rational function. $f(x)=\frac{x^2-x+2}{x-1}$

Solution:

First, notice that the graph is in two pieces. Most rational functions have graphs in multiple pieces. Find $y$-intercept by substituting zero for $x$ and solving for $y (f(x)): x=0→y=\frac{x^2-x+2}{x-1}=\frac{0^2-0+2}{0-1}=-2$,
$y$-intercept: $(0,-2)$
Asymptotes of $\frac{x^2-x+2}{x-1}$: Vertical: $x=1$, Slant asymptote: $y=x$
After finding the asymptotes, you can plug in some values for $x$ and solve for $y$. Here is the sketch for this function.

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Graphing Rational Expressions – Example 2:

Graph rational expressions. $f(x)=\frac{3x}{x^2-2x}$

Solution:

First, notice that the graph is in two pieces. Find $y$-intercept by substituting zero for $x$ and solving for $y (f(x)): x=0→y=\frac{3x}{x^2-2x}=\frac{3(0)}{(0^2-2(0)}=\frac{0}{0}$, $y$-intercept: None Asymptotes of $\frac{3x}{x^2-2x}$: vertical: $x=2$, Horizontal: $y=0$ After finding the asymptotes, you can plug in some values for $x$ and solve for $y$. Here is the sketch for this function.

Exercises for Graphing Rational Expressions

Graph these rational expressions.

• $\color{blue}{f(x)=\frac{x^2 -2x}{x-1}}$

• $\color{blue}{f(x)=\frac{x -5}{x^2-5x+1}}$

• $\color{blue}{f(x)=\frac{x^2}{4x-5}}$

• $\color{blue}{f(x)=\frac{5x-4}{2x^2-4x-5}}$

• $\color{blue}{f(x)=\frac{x^2 -2x}{x-1}}$

• $\color{blue}{f(x)=\frac{x -5}{x^2-5x+1}}$

• $\color{blue}{f(x)=\frac{x^2}{4x-5}}$

• $\color{blue}{f(x)=\frac{5x-4}{2x^2-4x-5}}$

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