1. Unit 3 – Rational Functions
Graph Characteristics
(Holes and Asymptotes)

2. RATIONAL FUNCTIONS A rational function is a function of the form:
where p and q are polynomials

3. What would the domain of a rational function be?
We’d need to make sure the denominator  0
Find the domain.
Factor to find “illegal” values!

4. The graph of looks like this:
If you choose x values close to 0, the graph gets close to the asymptote, but never touches it.
A dashed vertical line drawn at x = 0 is called a vertical asymptote. The line only signifies graph behavior near the excluded value. Vertical asymptotes exist when x-values make the denominator (not the numerator) = 0.

5. Let’s consider the graph
Behavior of x-values close to the excluded value x = 0.
As x-values get close to x = 0,
the y-values go to infinity

6. Does the function have an x intercept?
There is NOT a value that you can plug in for x that would make the function = 0. The graph approaches but never crosses the horizontal line y = 0. This is called a horizontal asymptote.
A graph will NEVER cross a vertical asymptote because the x value is “illegal” (would make the denominator 0)
A graph may cross a horizontal asymptote in the middle, but will eventually approach (not cross) the asymptote to the far right or left.

7. vertical translation,
moved up 3
Graph
This is just the parent function transformed. We can trade the terms places to make it easier to see this.
The vertical asymptote remains the same because in either function, x ≠ 0
The horizontal asymptote will move up 3 like the graph does.

8. Finding Asymptotes VERTICAL ASYMPTOTES
Vertical Asymptote -- any x-value that makes the denominator (not the numerator) = 0.
Factor top and bottom. Common factors -- holes.
Noncommon factors in denominator only – vertical asymptotes.
VERTICAL ASYMPTOTES
x = 4 and x = -1
are vertical asymptotes

9. Finding Asymptotes VERTICAL ASYMPTOTES
Vertical Asymptote -- any x-value that makes the denominator (not the numerator) = 0.
Factor top and bottom. Common factors -- holes.
Noncommon factors in denominator only – vertical asymptotes.
VERTICAL ASYMPTOTES

10. HORIZONTAL ASYMPTOTES
We compare the degrees of the polynomial in the numerator and the polynomial in the denominator to tell us about horizontal asymptotes.
1 < 2
degree of top = 1
If numerator degree < denominator degree, then a horizontal asymptote at y = 0. 1
degree of bottom = 2

11. HORIZONTAL ASYMPTOTES
Leading Coefficient --number in front of the highest-powered x term.
If the numerator degree = denominator degree, then a horizontal asymptote at:
y = leading coefficient of top
leading coefficient of bottom
degree of top = 2 1
degree of bottom = 2
horizontal asymptote at:

12. SLANT ASYMPTOTES
If the numerator degree > denominator degree, then no horizontal asymptote, instead a slant asymptote.
Slant asymptote -- complete long division and the quotient is the equation of the slant asymptote (ignore the remainder).
degree of top = 3
degree of bottom = 2
Oblique asymptote at y = x + 5

13. SUMMARY OF HOW TO FIND ASYMPTOTES
Vertical Asymptotes – factor numerator and denominator. Simplify common variable factors (holes). Then, set the denominator = 0 and solve any remaining variable factors.
To determine horizontal or oblique asymptotes, compare the degrees of the numerator and denominator.
If the degree of the top < the bottom, horizontal asymptote along the x axis (y = 0).
If the degree of the top = bottom, horizontal asymptote at y = leading coefficient of top over leading coefficient of bottom.
If the degree of the top > the bottom, slant asymptote found by long division.