# How to Graphs of Rational Functions?

You can graph rational functions in a few simple steps. Read this post to know more about 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.

The general form of a rational function is $$\frac{p(x)}{q(x)}$$. where $$p(x)$$ and $$q(x)$$ are polynomials and $$q(x)≠0$$.

## A step-by-step guide to graphs of rational functions

It can be challenging to draw graphs of rational functions. Finding the asymptotes and intercepts is a fine place to begin when attempting to graph a rational function.

Graphing rational functions involves a series of steps:

• Determine if the rational function has any asymptotes
• Asymptotes should be drawn as dotted lines.
• Find the rational function’s $$x$$ and $$y$$-intercepts, if any.
• Determine the $$y$$ values for a variety of $$x$$ values.
• Draw a smooth curve connecting the points by plotting them on a graph. Don’t cross-vertical asymptotes.

### Graphs of Rational Functions – Example 1:

Draw a graph of the following function. $$f(x)=\frac{3x+6}{x-1}$$

First, we need to determine the vertical asymptotes:

$$x-1=0 → x=1$$

And the horizontal asymptote is:

$$y=\frac{3}{1}=3$$

Then, the $$x$$-intercepts will be,

$$3x+6=0 → 3x=-6 → x=-\frac{6}{3}=-2$$ , the $$x$$-intercepts is $$(-2,0)$$

The $$y$$-intercept is:

$$f(0)=\frac{3(0)+6}{0-1}$$ → $$f(0)=\frac{6}{-1}=-6$$ , the $$y$$-intercepts is $$(0,-6)$$

We need to find more points on the function and graph the function.

Now, putting all this together gives the following graph.

## Exercises for Graphs of Rational Functions

### Draw a graph of the following function.

• $$\color{blue}{f(x)=\frac{9}{x^2-9}}$$
• $$\color{blue}{f(x)=\frac{4x^2+x}{2x^2+x}}$$
• $$\color{blue}{f(x)=\frac{x-2}{x^2-3x-4}}$$

• $$\color{blue}{f(x)=\frac{9}{x^2-9}}$$
• $$\color{blue}{f(x)=\frac{4x^2+x}{2x^2+x}}$$
• $$\color{blue}{f(x)=\frac{x-2}{x^2-3x-4}}$$

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