Rational FunctionsFinding Asymptotes

Rational Functions

What is a rational function? That’s right, a fraction!

What have we done with them already? Yep, we found the domain limits.▪ We set the denominator equal to “0”. That is

where it did not exist. Does the domain have anything to

do with asymptotes? Of course, that is the vertical asymptote.

Rational Functions

Vertical Asymptotes To find the vertical asymptote, set the

denominator equal to zero and solve. Find the vertical asymptotes: 6x2 – x – 2 = 0 x2 – x - 12 = 0 (x – 4)(x + 3) = 0 (x - (x + ) = 0 (3x – 2)(2x + 1) = 0 x = x = -

1.Set denominator equal

to 0.

2.Swing the 6 over.

3.Factor the polynomial.

4.Divide the 6 back out.

5.Solve.

Rational Functions

Horizontal Asymptotes When looking for the horizontal asymptote,

you must compare the degree of numerator to the degree of the denominator.▪ I will explain in a minute!

There will either be 1 asymptote or no asymptote.

There are 3 rules.▪ Equal▪ Big bootie▪ Big top

Rational Functions

Horizontal Asymptotes The degree is the largest exponent. You will need to

compare the top to the bottom. Here are the rules. Big Bootie▪ If the degree on bottom is bigger than the top, the HA is y

= 0. Big Top▪ If the degree on top is bigger than the bottom, there is no

HA. Equal (The tricky one)▪ If the degrees are the same, then the HA is a ratio of the

leading coefficients.

Rational Functions

Horizontal Asymptotes

y = 0

Lets find the horizontal asymptotes.1. What is the degree on top?

12. What is the degree on

bottom?2

3. Which is bigger top, bottom, or neither?Bottom

4. So, the HA is…….

Rational Functions

Horizontal Asymptotes

No HALets find the horizontal asymptotes.1. What is the degree on

top?3

2. What is the degree on bottom?2

3. Which is bigger top, bottom, or neither?Top

4. So, the HA is…….

Rational Functions

Horizontal Asymptotes

y = y = 2

Lets find the horizontal asymptotes.1. What is the degree on top?

22. What is the degree on

bottom?2

3. Which is bigger top, bottom, or neither?Neither

4. So, the HA is……A ratio of the leading coefficients

Rational Functions

Try it on your own. Find the vertical and horizontal

asymptotes for:

VA: x = 2HA: none

VA: x = 4 and x = 3HA: y = 1