1. Sullivan PreCalculus Section 3
Sullivan PreCalculus Section 3.4 Rational Functions II: Analyzing Graphs
Objectives
Analyze the Graph of a Rational Function
Solve Applied Problems Involving Rational Functions

2. To analyze the graph of a rational function:
a.) Find the Domain of the rational function.
b.) Locate the intercepts, if any, of the graph.
c.) Test for Symmetry. If R(-x) = R(x), there is symmetry with respect to the y-axis. If - R(x) = R(-x), there is symmetry with respect to the origin.
d.) Write R in lowest terms and find the real zeros of the denominator, which are the vertical asymptotes.
e.) Locate the horizontal or oblique asymptotes.
f.) Determine where the graph is above the x-axis and where the graph is below the x-axis.
g.) Use all found information to graph the function.

3. Example: Analyze the graph of

4. a.) x-intercept when x + 1 = 0: (-1,0)
b.) y-intercept when x = 0:
y - intercept: (0, 2/3)
c.) Test for Symmetry:
No symmetry

5. d.) Vertical asymptote: x = -3
Since the function isn’t defined at x = 3, there is a whole at that point.
e.) Horizontal asymptote: y = 2
f.) Divide the domain using the zeros and the vertical asymptotes. The intervals to test are:

6. Test at x = -4
Test at x = -2
Test at x = 1
R(-4) = 6
R(-2) = -2
R(1) = 1
Above x-axis
Below x-axis
Above x-axis
Point: (-4, 6)
Point: (-2, -2)
Point: (1, 1)
g.) Finally, graph the rational function R(x)

7. x = - 3
(-4, 6)
(1, 1)
(3, 4/3)
y = 2
(-2, -2)
(-1, 0)
(0, 2/3)

8. Example: The concentration C of a certain drug in a patients bloodstream t minutes after injection is given by:
a.) Find the horizontal asymptote of C(t)
Since the degree of the denomination is larger than the degree of the numerator, the horizontal asymptote of the graph of C is y = 0.

9. b.) What happens to the concentration of the drug as t (time) increases?
The horizontal asymptote at y = 0 suggests that the concentration of the drug will approach zero as time increases.
c.) Use a graphing utility to graph C(t). According the the graph, when is the concentration of the drug at a maximum?
The concentration will be at a maximum five minutes after injection.