Sullivan PreCalculusSection 3.4

Rational Functions II: Analyzing Graphs

Objectives

• Analyze the Graph of a Rational Function

• Solve Applied Problems Involving Rational Functions

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.

Example: Analyze the graph of 9

642)(

2

2

x

xxxR

R x

x x

x x( )

2 2 3

3 3

2

2 3 13 3

x xx x

2 1

33

xx

x,

Domain: x x x 3 3,

a.) x-intercept when x + 1 = 0: (-1,0)

b.) y-intercept when x = 0: 3

2

)30(

)10(2)0(

R

y - intercept: (0, 2/3)

3

12)(

x

xxR

c.) Test for Symmetry: R xx

x( )

( )( )

2 13

)()()( xRxRxR No symmetry

R xx

xx( ) ,

2 1

33

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:

x x x3 3 1 1

x x x3 3 1 1

Test at x = -4

R(-4) = 6

Above x-axis

Point: (-4, 6)

Test at x = -2

R(-2) = -2

Below x-axis

Point: (-2, -2)

Test at x = 1

R(1) = 1

Above x-axis

Point: (1, 1)

g.) Finally, graph the rational function R(x)

8 6 4 2 0 2 4 6

10

5

5

10

(-4, 6)

(-2, -2) (-1, 0) (0, 2/3)

(1, 1) (3, 4/3)

y = 2

x = - 3

Example: The concentration C of a certain drug in a patients bloodstream t minutes after injection is given by:

)25(

50)(

2

t

ttC

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.

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.