1. Discussion
X-intercepts

2. X-intercepts of Rational Function
Notes
X-intercepts of Rational Function
To find the x-int of Rational Functions, set the numerator equal to zero and solve for x.

3. Find all x-intercepts of each function.
Practice
Find all x-intercepts of each function.

4. Discussion
y-intercepts

5. y-intercepts of Rational Function
Notes
y-intercepts of Rational Function
To find the y-int of Rational Functions, substitute 0 for x.

6. Find all y-intercepts of each function.
Practice
Find all y-intercepts of each function.

7. Discussion Domain and Range What is the domain of this function?
Are there any numbers that x is not allowed to equal?

8. The domain is “split” by the vertical asymptotes!
Practice
Find the Domain.
The domain is “split” by the vertical asymptotes!

9. Notes
Asymptote
An asymptote is a line that a function approaches but never actually reaches.
Vertical Asymptote
Horizontal Asymptote

10. Notes Vertical Asymptotes
A rational function has a vertical asymptote at each value of x that makes only the denominator equal zero.
Example:
Vertical Asymptotes:

11. Find the Vertical Asymptotes:
Practice
Find the Vertical Asymptotes:

12. Horizontal Asymptotes
Notes
Horizontal Asymptotes
a: leading coefficient of the numerator
b: leading coefficient of the denominator
n: degree of the numerator
m: degree of the denominator
If m = n, then y = a/b is an asymptote
If n < m, then y = 0 is an asymptote
If n > m, then there is no horizontal asymptote

13. Find the Horizontal Asymptotes:
Practice
Find the Horizontal Asymptotes:

14. Discussion
End Behavior

15. Notes
End Behavior
The shape of a fucntion when x approaches positive or negative infinity.
Where does the graph “level out?”
The horizontal asymptote is sometimes called the “end-behavior asymptote.” Why?

16. State the end behavior of each function.
Practice
State the end behavior of each function.

17. Notes
Holes
If a value of x makes both the numerator and denominator of the fraction equal zero, then the function has a hole at that point.
Example:
Hole:

18. Practice
Find the Holes:

19. Notes
Slant Asymptote
A rational Function will have a slant asymptote if the degree of the top is exactly one more than the degree of the bottom.
Example:

20. Determine if the function has a slant asymptote.
Practice
Determine if the function has a slant asymptote.

21. Find the equation of the slant asymptote:
Discussion
Find the equation of the slant asymptote:

22. Notes
Slant Asymptote
To find the slant asymptote of a rational function, divide the top by the bottom. The equation of the asymptote is the quotient (ignore the remainder).

23. Find the slant asymptote.
Practice
Find the slant asymptote.

24. Discussion
Graph the function.
x-int:
V.A.:
H.A.:
S.A.:
Holes:

25. Practice
Graph each function.

26. Class Work (30 minutes)
Find the HOLES: Factor both numerator and denominator and look for common factors. If there are any, set them equal to zero and solve. Those are your holes. If no factors repeat, there are no holes.
Find the X-INTERCEPT(S): Set your numerator equal to zero and solve
Find the Y-INTERCEPT: Plug in zero for all x-values
Find the VERTICAL ASYMPTOTE: Set your denominator equal to zero and solve. If there is a hole, there is no V.A.
Find your HORIZONTAL ASYMPTOTE: If the degree of the numerator is equal to the degree of the denominator, take the coefficient of the terms. If the degree of the numerator is larger than the denominator, there is no H.A, there is a SLANT ASYMPTOTE. If the degree of the numerator is less than the denominator then the H.A is the line y=0.

27. Inverses of Rational Functions
Cross multiply
Solve for x (factor)
Switch x and y
Rewrite as an inverse
Example:
Class Work (30 minutes)