2.6: Rational Functions

Rational Functions


A rational function is a function that can be written as

Where N(x) and D(x) are polynomials.

• Example of a rational function:

A function likeis considered a rational function, because it can be rewritten as

*NOTE: When you divide by a fraction, the answer gets larger.

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Vertical Asymptotes

Consider the equation .

The denominator cannot be 0, so x ≠ 3.

 Notice that the graph avoids x = 3, and



There can be more than one vertical asymptote, as is demonstrated in the equation, because x = ± 1.

 

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Horizontal Asymptotes

Unlike vertical asymptotes, there can only be one horizontal asymptote.

In order to determine what the horizontal asymptote is

• Pretend that x is a large number like 1,000,000,000
• Consider the highest degree in the numerator and denominator

Because x is so large, everything but the highest degree polynomial is irrelevant. There are three different horizontal asymptote scenarios.

1. Higher degree polynomial in the denominator

 

 Look only at the highest degree polynomials, 3x and 2x²

, so the H.A. is 0.

2. Highest degree polynomial same in both numerator and denominator

Look at only the highest degree polynomials, 3x² and 2x². The x²s cancel each other out, and , so the H.A. is 3/2.

3. Higher degree polynomial in the numerator

It does not level off, so and there is NO H.A.

 Still confused? CLICK HERE to watch a video explaining rational function asymptotes.