Graphing rational functions—overview

By Tutor GuyNo Comments

 

Graphing rational functions can be scary for a lot of students because there are so many details to manage. But you can make the process simpler by breaking down all the requirements into small steps. In this post I list all of the steps you should follow to analyze a rational function. Note that you can do these steps in any order. In subsequent posts, I will walk you through each of the steps and show you how to execute them.

Let’s start with the definition of a rational function. A rational function is a function that consists of a fraction, where both the numerator and denominator are polynomials. Or in symbols, if g(x) and h(x) are polynomials, then

f(x)=\dfrac{g(x)}{h(x)}

is called a rational function.

 

To analyze and graph a rational function, you need to do all of the following steps:

  • Find the y-intercept (if it exists)
  • Find the x-intercept(s) (if they exist)
  • Determine the location of holes, if any
  • Find the vertical asymptotes, if any
  • Find the horizontal or oblique asymptotes, if any
  • Determine the end behavior (if no horizontal or oblique asymptotes)

[Note that this is a list for algebra 2 and precalc students. When you get to calculus, there will be additional steps to completely analyze the function.]

This probably feels like a long and complicated list, but for the most part, each step by itself is not very difficult. Just perform each step in turn and you can become an expert at rational functions. Check out my other posts in this section that describe how to perform each of these steps.

Algebra 2, Precalc/Trig
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