1. Rational Functions

2. A rational function is a function of the form
Where p and q are polynomial functions and q is not the zero polynomial. The domain consists of all real numbers except those for which the denominator is 0.

3. Find the domain of the following functions:
All real numbers except -6 and-2.
All real numbers except -4 and 4.
All real numbers.

4. Behavior Near Undefined Values
Examine the behavior of the function near the undefined values and as x goes to  (as x goes to the extremes.

6. Vertical Asymptotes
x = c y x
x = c y x

7. Theorem Locating Vertical Asymptotes
A rational function,
in lowest terms, will have a vertical asymptote x = r, if x - r is a factor of the denominator q.

8. Vertical asymptotes: x = -1 and x = 1
Find the vertical asymptotes, if any, of the graph of each rational function.
Vertical asymptotes: x = -1 and x = 1
No vertical asymptotes
Vertical asymptote: x = -4

9. y
Horizontal Asymptotes
y = R(x)
y = L x y
y = L x
y = R(x)

10. If an asymptote is neither horizontal nor vertical it is called oblique.x

11. Consider the rational function
If n < m, then y = 0 is a horizontal asymptote of the graph of R.
If n = m, then y = an / bm is a horizontal asymptote of the graph of R.
If n = m + 1, then y = ax + b is an oblique asymptote of the graph of R. Found using long division.
4. If n > m + 1, the graph of R has neither a horizontal nor oblique asymptote. End behavior found using long division.

12. Find the horizontal and oblique asymptotes if any, of the graph of
Horizontal asymptote: y = 0
Horizontal asymptote: y = 2/3

13. Oblique asymptote: y = x + 6