Rational Functions

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8-4
Rational Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Find the zeros of each function.
1. f(x) x2 2x 15
5, 3
2. f(x) x2 49
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Simplify. Identify any x-values for which the
expression is undefined.
3.
x ? 1
4.
x ? 6
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Objectives
Graph rational functions. Transform rational
functions by changing parameters.
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Vocabulary
rational function discontinuous
function continuous function hole (in a 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)
Because h 2, translate f 2 units left.
Because k 3, translate f 3 units down.
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Check It Out! Example 1
a. g(x)
b. g(x)
Because k 1, translate f 1 unit up.
Because h 4, 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 2 Determining Properties of Hyperbolas
Identify the asymptotes, domain, and range of the
function g(x) 2.
h 3, k 2.
Vertical asymptote x 3
The value of h is 3.
Domain xx ? 3
Horizontal asymptote y 2
The value of k is 2.
Range yy ? 2
Check Graph the function on a graphing
calculator. The graph suggests that the function
has asymptotes at x 3 and y 2.
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Check It Out! Example 2
Identify the asymptotes, domain, and range of the
function g(x) 5.
h 3, k 5.
Vertical asymptote x 3
The value of h is 3.
Domain xx ? 3
Horizontal asymptote y 5
The value of k is 5.
Range yy ? 5
Check Graph the function on a graphing
calculator. The graph suggests that the function
has asymptotes at x 3 and y 5.
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A discontinuous function is a function whose
graph has one or more gaps or breaks. The
hyperbola graphed in Example 2 and many other
rational functions are discontinuous functions.
A continuous function is a function whose graph
has no gaps or breaks. The functions you have
studied before this, including linear, quadratic,
polynomial, exponential, and logarithmic
functions, are continuous functions.
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The numerator of this function is 0 when x 3
or x 2. Therefore, the function has
x-intercepts at 2 and 3. The denominator of this
function is 0 when x 1. As a result, the
graph of the function has a vertical asymptote at
the line x 1.
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Example 3 Graphing Rational Functions
with Vertical Asymptotes
Identify the zeros and vertical asymptotes of
f(x) .
Step 1 Find the zeros and vertical asymptotes.
Factor the numerator.
The numerator is 0 when x 4 or x 1.
Zeros 4 and 1
The denominator is 0 when x 3.
Vertical asymptote x 3
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Example 3 Continued
Step 2 Graph the function.
Plot the zeros and draw the asymptote. Then make
a table of values to fill in missing points.
x 8 4 3.5 2.5 0 1 4
y 7.2 0 4.5 10.5 1.3 0 3.4
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Check It Out! Example 3
Identify the zeros and vertical asymptotes of
f(x) .
Step 1 Find the zeros and vertical asymptotes.
Factor the numerator.
The numerator is 0 when x 6 or x 1 .
Zeros 6 and 1
The denominator is 0 when x 3.
Vertical asymptote x 3
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Check It Out! Example 3 Continued
Step 2 Graph the function.
Plot the zeros and draw the asymptote. Then make
a table of values to fill in missing points.
Vertical asymptote x 3
x 7 5 2 1 2 3 7
y 1.5 2 4 0 4.8 6 10.4
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Some rational functions, including those whose
graphs are hyperbolas, have a horizontal
asymptote. The existence and location of a
horizontal asymptote depends on the degrees of
the polynomials that make up the rational
function.
Note that the graph of a rational function can
sometimes cross a horizontal asymptote. However,
the graph will approach the asymptote when x is
large.
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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 with a graphing calculator or by 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 with a graphing calculator or by using a
table of values.
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Example 4C 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 3.
Zero 3
The denominator is 0 when x 1.
Vertical asymptote x 1
Horizontal asymptote y 4
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Example 4C Continued
Identify the zeros and asymptotes of the
function. Then graph.
Graph with a graphing calculator or by using a
table of values.
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Check It Out! Example 4a
Identify the zeros and asymptotes of the
function. Then graph.
f(x)
Factor the numerator.
The numerator is 0 when x 3 or x 5.
Zeros 3 and 5
The denominator is 0 when x 1.
Vertical asymptote x 1
Horizontal asymptote none
Degree of p gt degree of q.
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Check It Out! Example 4a Continued
Identify the zeros and asymptotes of the
function. Then graph.
Graph with a graphing calculator or by using a
table of values.
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Check It Out! Example 4b
Identify the zeros and asymptotes of the
function. Then graph.
Factor the denominator.
The numerator is 0 when x 2.
Zero 2
Vertical asymptote x 1, x 0
The denominator is 0 when x 1 or x 0.
Horizontal asymptote y 0
Degree of p lt degree of q.
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Check It Out! Example 4b Continued
Identify the zeros and asymptotes of the
function. Then graph.
Graph with a graphing calculator or by using a
table of values.
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Check It Out! Example 4c
Identify the zeros and asymptotes of the
function. Then graph.
Factor the numerator and the denominator.
The denominator is 0 when x 3.
Vertical asymptote x 3, x 3
Horizontal asymptote y 3
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Check It Out! Example 4c Continued
Identify the zeros and asymptotes of the
function. Then graph.
Graph with a graphing calculator or by using a
table of values.
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In some cases, both the numerator and the
denominator of a rational function will equal 0
for a particular value of x. As a result, the
function will be undefined at this x-value. If
this is the case, the graph of the function may
have a hole. A hole is an omitted point in a
graph.
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Example 5 Graphing Rational Functions with Holes
Factor the numerator.
There is a hole in the graph at x 3.
The expression x 3 is a factor of both the
numerator and the denominator.
Divide out common factors.
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Example 5 Continued
The graph of f is the same as the graph of y x
3, except for the hole at x 3. On the graph,
indicate the hole with an open circle. The domain
of f is xx ? 3.
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Check It Out! Example 5
Factor the numerator.
There is a hole in the graph at x 2.
The expression x 2 is a factor of both the
numerator and the denominator.
Divide out common factors.
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Check It Out! Example 5 Continued
The graph of f is the same as the graph of y x
3, except for the hole at x 2. On the graph,
indicate the hole with an open circle. The domain
of f is xx ? 2.
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Lesson Quiz Part I
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.
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asymptotes x 1, y 2 Dxx ? 1 Ryy ?
2
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Lesson Quiz Part II
3. Identify the zeros, asymptotes, and holes in
the graph of . Then
graph.
zero 2 asymptotes x 0, y 1 hole at x 1