3.6 Warm UpFind the initial point, state the

domain & range, and compare to the parent function f(x) = √x.

1. y = 3√x – 1

2. y = -1/2√x

3. y = - √(x-1) + 2

4. y = 3√-x

3.6 Graph Rational Functions

Vocabulary:

A rational function has a rule given by a fraction whose numerator and denominator are polynomials and whose denominator is not 0.

An asymptote is a line that a graph approaches more and more closely.

EXAMPLE 1 Compare graph of y = with graph of y = ax

1x

b. The graph of1

y = 3xvertical stretch with a reflection in the x - axis of the graph of

vertical shrink of the graph of

=y1x y = x

1

–2x is aThe graph ofa.

Compare to parent function:

1.

2.

3.

6y

x

1

6y

x

3

2y

x

Steps to graph: Identify and sketch asymptotes. Make a t-chart w/ three points on

either side of the vertical asymptote. Plot points & draw graph. Compare to the parent function. Identify domain and range.

EXAMPLE 2

Graph y = + 3 and identify its domain and range. Compare the graph with the graph of y = .

1x 1

x

Graph y = + k1x

EXAMPLE 3

Graph y = and identify its domain and range.

Compare the graph with the graph of y = .

1x– 2

1x

Graph y = 1

x – h

EXAMPLE 4 Graph y = + k ax – h

Graph y = – 3. 2

x + 1Identify the domain and range and compare with the parent function.

GUIDED PRACTICE for Example 4

5. Graph y = + 6 4x – 5

Identify the domain and range and compare with the parent function.

GUIDED PRACTICE for Example 4

6. For which function is the domain all real numbers except –3 and the range all real numbers except 7 ?

A. y = + 7.2x – 3 B. y = – 7.2

x – 3

C. y = + 7.2x + 3 D. y = – 7.2

x + 3

ANSWER

C. y = + 7.2x + 3

Determine the asymptotes of the graph of the function.

1.

2.

714

5y

x

103

5 5y

x

Write an equation whose graph has the given asymptotes and passes through the given point.

1. x = -2, y = 5; (0, -2)

Write an equation whose graph has the given asymptotes and passes through the given point.

2. x = -4, y = -4; (-5, 3)