How to Graph Rational Expressions

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.

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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.

  • \(\color{blue}{f(x)=\frac{x^2 -2x}{x-1}}\)
  • \(\color{blue}{f(x)=\frac{x -5}{x^2-5x+1}}\)
  • \(\color{blue}{f(x)=\frac{x^2}{4x-5}}\)
  • \(\color{blue}{f(x)=\frac{5x-4}{2x^2-4x-5}}\)
  • \(\color{blue}{f(x)=\frac{x^2 -2x}{x-1}}\)
This image has an empty alt attribute; its file name is Graphing-Rational-Expressions-Example-4.png
  • \(\color{blue}{f(x)=\frac{x -5}{x^2-5x+1}}\)
This image has an empty alt attribute; its file name is Graphing-Rational-Expressions-Example-5.png
  • \(\color{blue}{f(x)=\frac{x^2}{4x-5}}\)
This image has an empty alt attribute; its file name is Graphing-Rational-Expressions-Example-6.png
  • \(\color{blue}{f(x)=\frac{5x-4}{2x^2-4x-5}}\)
This image has an empty alt attribute; its file name is Graphing-Rational-Expressions-Example-7.png

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