1. Finding Asymptotes
Rational Functions

2. 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.

3. Rational Functions Vertical Asymptotes
To find the vertical asymptote, set the denominator equal to zero and solve.
Find the vertical asymptotes: 𝑓 𝑥 = 5𝑥+2 6 𝑥 2 −𝑥 −2
6x2 – x – 2 = 0
x2 – x - 12 = 0
(x – 4)(x + 3) = 0
(x )(x ) = 0
(3x – 2)(2x + 1) = 0
x = x =
Set denominator equal to 0.
Swing the 6 over.
Factor the polynomial.
Divide the 6 back out.
Solve.

4. 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

5. 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.

6. Rational Functions Horizontal Asymptotes 𝑓 𝑥 = 8𝑥+3 4 𝑥 2 +1 y = 0
𝑓 𝑥 = 8𝑥+3 4 𝑥 2 +1
y = 0
Lets find the horizontal asymptotes.
What is the degree on top? 1
What is the degree on bottom? 2
Which is bigger top, bottom, or neither?
Bottom
So, the HA is…….

7. Rational Functions Horizontal Asymptotes 𝑓 𝑥 = 8 𝑥 3 +3 4 𝑥 2 +1 No HA
𝑓 𝑥 = 8 𝑥 𝑥 2 +1
No HA
Lets find the horizontal asymptotes.
What is the degree on top? 3
What is the degree on bottom? 2
Which is bigger top, bottom, or neither?
Top
So, the HA is…….

8. Rational Functions Horizontal Asymptotes 𝑓 𝑥 = 8 𝑥 2 +3 4 𝑥 2 +1
𝑓 𝑥 = 8 𝑥 𝑥 2 +1
y = 8 4 = 2 1
y = 2
Lets find the horizontal asymptotes.
What is the degree on top? 2
What is the degree on bottom?
Which is bigger top, bottom, or neither?
Neither
So, the HA is……
A ratio of the leading coefficients

9. Rational Functions Try it on your own.
Find the vertical and horizontal asymptotes for:
𝑓 𝑥 = 𝑥 2 +3𝑥+2 𝑥 −2
𝑓 𝑥 = 𝑥 2 −4𝑥 𝑥 2 −7𝑥+12
VA: x = 2
HA: none
VA: x = 4 and x = 3
HA: y = 1