Sep 01

Graphing rational functions—4. Drawing the graph

By Tutor GuyNo Comments

 

In previous posts, I described the steps you follow to analyze a rational function. If you follow these steps, you can be a rational function superstar too. In this post, I put all the pieces together to show how you use the information you’ve obtained to plot the graph. [For details on how to execute the various steps, please see other posts on this website.]

Example: Graph the following rational function.

f(x)=\dfrac{2x^3-16x^2+38x-24}{x^3-4x^2+x+6}

Solution: We break this into many steps.

1. Find the y-intercept:

f(0)=\dfrac{-24}{6}=-4

2. Fully factor the function:

f(x)=\dfrac{2x^3-16x^2+38x-24}{x^3-4x^2+x+6}=\dfrac{2(x-1)(x-3)(x-4)}{(x+1)(x-2)(x-3)}

 3. Find the vertical asymptotes, if any:

The vertical asymptotes are at x=-1 and x=2.

 

4.Find the x-intercept(s), if any:

f(x)=0 @ x=1 and x=4

 

5. Find the hole(s), if any:

There is a hole at the point (3,-1)

 

6. Find the horizontal or oblique asymptote, if any, or characterize the end behavior:

The horizontal asymptote is at y=2

 

7. Place all of this information on the graph:

 

 

 

 

 

 

 

 

 

 

 

 8. Use all the information you have plotted to complete the graph of the function:

Algebra 2, Precalc/Trig
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