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.

### 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.

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

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

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

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

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

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

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

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