1. 3.7 Graphs of Rational Functions
Objectives: Graph rational functions. Determine vertical, horizontal, and slant asymptotes.

2. Horizontal Asymptote:
A line that a graph approaches but never intersects.
Vertical Asymptote:
The line x=a is a vertical asymptote for a function f(x) if f(x)→ ∞ or f(x) → -∞ as
x → a from either the left or the right .
Horizontal Asymptote:
The line y=b is a horizontal asymptote for a function f(x) if f(x)→ b as x → ∞ as
x → - ∞.
See example 1 to see methods of how to find horizontal and vertical asymptotes.

3. Ex. 1)
Determine the asymptotes for the graph of
f(x) = x/(x-1)
Ex. 2)
Use the parent graph f(x)=1/x to graph each function. Describe the transformation(s) that take place. Identify the new location of each asymptote.
a.) g(x)=1/(x-1)
b.) h(x)=-2/x
c.) k(x) = 7/(x + 5)
d.) m(x) = 1/(x – 3) + 2

4. Slant asymptote:
The oblique line l is a slant asymptote for a function f(x) if the graph of f(x) approaches l as x→∞ or as x →-∞.
(Top heavy rational expression.)
-bottom heavy → horizontal at y= 0
See example 3 in book on pg. 183.
Ex. 4)
Determine the slant asymptote for
f(x)=(4x² + 6x – 37) / (x + 4)

5. Point Discontinuity:
→hole in graph
-when numerator and denominator
contain a common linear factor.
Ex. 5)
Graph y = (x + 2)(x – 3)
x(x – 4)²(x + 2)