1. Chapter 4: Polynomial & Rational Functions 4.4: Rational Functions
Essential Question: How can you determine the vertical and horizontal asymptotes of an equation?

2. 4.4: Rational Functions Domain of Rational Functions
The domain is the set of all real numbers that are not zeros of its denominator.
Example 1: The Domain of a Rational Function
Find the domain of each rational function
All real numbers except x = 0
All real numbers except x2 – x – 6 = 0
(x – 3)(x + 2) = 0, so all real numbers except for 3 and -2

3. 4.4: Rational Functions Properties of Rational Graphs Example
Intercepts
As with any graph, the y-intercept is at f(0)
The x-intercepts are when the numerator = 0 and the denominator does not equal 0.
Example
Find the intercepts of
y-intercept:
x-intercepts: Neither are solutions of x – 1 = so both are x-intercepts

4. 4.4: Rational Functions Properties of Rational Graphs (continued)
Vertical Asymptotes
Whenever only the denominator = 0 (numerator is not 0)
Vertical asymptotes will either spike up to ∞ or down to -∞
Big-Little Concept
Dividing by a small number results in a large number
Dividing by a large number results in a small number.

5. 4.4: Rational Functions Behavior near a Vertical Asymptote
Describe the graph of near x = 2
As x gets closer and closer to 2 from the right (2.1, 2.01, 2.001, …) the denominator becomes a really small positive number.
Division by a small positive number means the graph of f(x) approaches ∞ from the right
As x gets closer and closer to 2 from the left (1.9, 1.99, , …) the denominator becomes a really small negative number
Division by a small negative number means the graph of f(x) approaches -∞ from the left

6. 4.4: Rational Functions Holes
When a number c is a zero of both the numerator and denominator of a rational function, the function might have a vertical asymptote, or it might have a hole.
Example #1
But this is not the same as the function g(x) = x + 2, as f(2) = while g(2) = 4, so though they may look the same, f(x) has a hole at x = 2
Example #2
The graph of x2/x3 looks the same as 1/x, and has a vertical asymptote, as neither of the functions are defined at x = 0

7. 4.4: Rational Functions Holes
If and a number d exists such that g(d) and h(d) = 0
If the degree of the numerator is greater than (or equal to) the degree of the denominator after simplification, then the function has a hole at x = d
= x = 5
If the degree of the denominator is greater than the degree of the numerator after simplification, then the function has a vertical asymptote at x = d
= x = 5
It is far easier to first determine the domain of the function, and then visually inspect to see whether “hiccups” in the domain are holes or asymptotes.

8. 4.4: Rational Functions End Behavior (Horizontal Asymptotes)
The horizontal asymptote is found by determining what the function will be when x is extraordinarily large
When x is large, a polynomial function behaves like its highest degree term
Example #1
List the vertical asymptotes and describe the end behavior.
There is a vertical asymptote at x = 5/2
Both numerator and denominator have the same degree, so the horizontal asymptote is at y = -3/2

9. 4.4: Rational Functions Asymptotes, Example #2 Vertical asymptotes at:
(x – 2)(x + 2) = 0
Vertical asymptotes at x = 2 or x = -2
Horizontal Asymptotes at:
As x becomes large, 1/x becomes small.
So horizontal asymptote at y = 0.

10. 4.4: Rational Functions Asymptotes, Example #3 Vertical asymptotes at:
Graphing tells you there is only one root, at x = -1
Vertical asymptotes at x = -1
Horizontal Asymptotes at:
Horizontal asymptote at y = 2.

11. 4.4: Rational Functions Assignment Page 290 1 – 49, odd problems
Due tomorrow
Show work
Only worry about holes, vertical and horizontal asymptotes.
Disregard stuff on slant/parabolic asymptotes
Ignore the graphing (you have a graphing calculator for that)