Notes:

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Notes
1. Using the graph of f(x) as a guide,
describe the transformation and graph the
function g(x) .
3. Identify the zeros and asymptotes of the
function. Then graph.
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Example 1 Transforming Rational Functions
Using the graph of f(x) as a guide,
describe the transformation and graph each
function.
A. g(x)
B. g(x)
translate f 2 units left.
translate f 3 units down.
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Example 2
a. g(x)
b. g(x)
translate f 1 unit up.
translate f 4 units left.
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The values of h and k affect the locations of
the asymptotes, the domain, and the range of
rational functions whose graphs are hyperbolas.
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Example 3 Determining Properties of Hyperbolas
Identify the asymptotes, domain, and range of the
function g(x) 2.
Vertical asymptote x 3
Domain xx ? 3
Horizontal asymptote y 2
Range yy ? 2
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Example 4A Graphing Rational Functions with
Vertical and Horizontal Asymptotes
Identify the zeros and asymptotes of the
function. Then graph.
Factor the numerator.
The numerator is 0 when x 4 or x 1.
Zeros 4 and 1
The denominator is 0 when x 0.
Vertical asymptote x 0
Horizontal asymptote none
Degree of p gt degree of q.
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Example 4A Continued
Identify the zeros and asymptotes of the
function. Then graph.
Graph using a table of values.
Vertical asymptote x 0
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Example 4B Graphing Rational Functions with
Vertical and Horizontal Asymptotes
Identify the zeros and asymptotes of the
function. Then graph.
Factor the denominator.
The numerator is 0 when x 2.
Zero 2
The denominator is 0 when x 1.
Vertical asymptote x 1, x 1
Horizontal asymptote y 0
Degree of p lt degree of q.
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Example 4B Continued
Identify the zeros and asymptotes of the
function. Then graph.
Graph using a table of values.
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Notes
1. Using the graph of f(x) as a guide,
describe the transformation and graph the
function g(x) .
g is f translated 4 units right.
2.
asymptotes x 1, y 2 Dxx ? 1 Ryy ?
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Notes 3
3. Identify the zeros and asymptotes of the
function. Then graph.
Factor the numerator.
The numerator is 0 when x 3.
Zero 3
The denominator is 0 when x 1.
Vertical asymptote x 1
Horizontal asymptote y 4
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Notes 3 Continued
Identify the zeros and asymptotes of the
function. Then graph.
Graph using a table of values.