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What you will learn
1. How to graph a rational function based on the parent graph.
2. How to find the horizontal, vertical and slant asymptotes for a rational function.
Objective: Section 3-7 Graphs of Rational Functions
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Yeah! Definitions
1. Rational Function: A quotient of two polynomial functions.
2. Asymptote: A line that a graph approaches but never intersects. (Can be horizontal, vertical, or slant)
Objective: Section 3-7 Graphs of Rational Functions
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Types of Asymptotes Horizontal asymptote: the line y = b is a
horizontal asymptote for a function f(x) if f(x) approaches b as x approaches infinity or as x approaches negative infinity.
Vertical asymptote: the line x = a is a vertical asymptote for a function f(x) if f(x) approaches infinity or f(x) approaches negative infinity as x approaches “a” from either the left or the right.
Slant asymptote: the oblique line “l” is a slant asymptote for a function f(x) if the graph of y = f(x) approaches “l” as x approaches infinity or as x approaches negative infinity.
Objective: Section 3-7 Graphs of Rational Functions
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Visual Vocabulary
x-10 -5 5 10
y
-10
-5
5
10
Vertical asymptote
Horizontal Asymptote
xy
1
Objective: Section 3-7 Graphs of Rational Functions
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Slant Asymptote
x-10 -5 5 10
y
-10
-5
5
10
Slant Asymptote
Objective: Section 3-7 Graphs of Rational Functions
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Finding Asymptotes
Find the asymptotes for the graph of
Vertical asymptote: value of x that causes a “0” in the denominator. x – 2 = 0 x = 2 is vert. as.
Check:
2
13)(
x
xxf
X F(x)
1.9
1.99
1.999
1.9999
Objective: Section 3-7 Graphs of Rational Functions
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Finding Asymptotes (cont.)Find the asymptotes for the graph of
Horizontal asymptotes: Divide the numerator and the denominator by the highest power of x. Ask yourself, as x gets infinitely large, what would the value of the function be?
2
13)(
x
xxf
Objective: Section 3-7 Graphs of Rational Functions
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You Try
Determine the asymptotes for the graph of:
1)(
x
xxf
Objective: Section 3-7 Graphs of Rational Functions
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Finding Slant Asymptotes Slant asymptotes occur when the degree of the numerator of
a rational function is exactly one greater than the degree of the denominator.
Example: Find the slant asymptote for:
2
132)(
2
x
xxxf
Objective: Section 3-7 Graphs of Rational Functions
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You Try Find the slant asymptote for:
4
3764)(
2
x
xxxf
Objective: Section 3-7 Graphs of Rational Functions
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Graphing Rational Functions Can you predict what will happen as we graph
the following:
1. 2.
3. 4.
5
1)(
x
xg xxh
2
1)(
3
4)(
x
xk 42
6)(
x
xf
Objective: Section 3-7 Graphs of Rational Functions
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Let’s See
5
1)(
x
xg
x-10 -5 5 10
y
-10
-5
5
10
Objective: Section 3-7 Graphs of Rational Functions
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How About…
xxh
2
1)(
x-10 -5 5 10
y
-10
-5
5
10
Objective: Section 3-7 Graphs of Rational Functions
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How About…
3
4)(
x
xk
x-10 -5 5 10
y
-10
-5
5
10
Objective: Section 3-7 Graphs of Rational Functions
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How About
42
6)(
x
xf
x-10 -5 5 10
y
-10
-5
5
10
Objective: Section 3-7 Graphs of Rational Functions
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You Try! How will the following functions relate to the
parent graph?
23
1)(
5
7)(
2)(
1
1)(
xxm
xxk
xxh
xxg
Objective: Section 3-7 Graphs of Rational Functions
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A Last Note… Graph
)2)(3(
)1)(3(
xxx
xxy
Objective: Section 3-7 Graphs of Rational Functions
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Homework Page 186, 14, 16, 18, 22, 24, 26, 30, 32,
36
