# How to Graph Rational Expressions

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

## 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.

### 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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