Sullivan Algebra and Trigonometry: Section 4.4

Rational Functions II: Analyzing Graphs

Objectives

• Analyze the Graph of a Rational Function

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.

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

)()( xRxR 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 hole 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) There is a HOLE at this Point.

y = 2

x = - 3