How to Graph Rational Functions?
The rational function is defined as a polynomial coefficient whose denominator has a degree of at least 1. In other words, there must be a variable in the denominator. You can graph Rational Functions in a few simple steps. Join us to learn more about graphing Rational Functions.

The general form of a rational function is \(\frac {p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials and \(q(x) ≠ 0\).
Related Topics
- How to Graph Rational Expressions
- How to Add and Subtract Rational Expressions
- How to Multiply Rational Expressions
- How to Solve Rational Equations
A step-by-step guide to graphing rational functions
Each rational function may be graphed using the following steps:
- Find the \(y\)-intercept by evaluating the function at zero.
- Multiply the numerator and denominator together.
- Determine where each numerator component is zero to discover the \(x\)-intercepts for factors in the numerator that are not common to the denominator.
- Determine the behavior of the graph at those places by finding the multiplicities of the \(x\)-intercepts.
- Note the multiplicities of the zeros in the denominator to determine the local behavior. Find the vertical asymptotes for those components that are not common to the numerator by setting them to zero and then solving them.
- Find the removable discontinuities in the denominator that are similar to factors in the numerator by setting those factors to \(0\) and then solving.
- To get the horizontal or slant asymptotes, compare the degree of the numerator and denominator.
- Draw the graph.
Graphing Rational Functions – Example 1:
Sketch the graph of the following function: \(f(?)=\frac{3?+6}{?−1}\)
First, find the \(y\)-intercept: \(f(0)=\frac{3(0)+6}{0−1}=\frac {6}{-1}=-6 → (0,-6)\)
Then, find the \(x\)-intercepts: \(3x+6=0 → 3x=-6 → x=\frac{-6}{3}=-2 → (-2,0)\)
Now, we have to determine the asymptotes:
Vertical asymptote: \(x=1\), Horizontal asymptote: \(?=3\)
Now, we only need points in each region of \(x\)’s. Since \(y\)-intercept and \(x\)-intercept is currently in the left region, we will not need to point there. This means that we need to get a point in the right region. It does not matter how much \(x\) we choose here, we just need to keep it relatively small to fit on our graph.
\(f(2)=\frac{3(2)+6}{2−1}=\frac{12}{1}=12\)
By putting all this together, the following diagram is obtained.

Exercises for Graphing Rational Functions
Graph these rational functions.
- \(\color{blue}{f(x)=\frac{x^2 -3x}{x-5}}\)

- \(\color{blue}{f(x)=\frac{x-6}{x^2-5x+1}}\)

- \(\color{blue}{f(x)=\frac{x^2}{12x+6}}\)



- \(\color{blue}{f(x)=\frac{x^2 -3x}{x-5}}\)




Related to This Article
More math articles
- A Comprehensive Collection of Free HiSET Practice Tests
- A Complete Explanation of the Continuity over an Interval
- Overview of the CBEST Mathematics Test
- Representing Percentage
- How to Solve Natural Logarithms? (+FREE Worksheet!)
- Math Café: How to Learn the Art of Writing and Solving Two-variable Equations
- How to Write an Equation from a Graph?
- 5th Grade ILEARN Math Worksheets: FREE & Printable
- How to Compare Decimals? (+FREE Worksheet!)
- 3rd Grade MEAP Math Practice Test Questions
What people say about "How to Graph Rational Functions? - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.