Rational Functions and Their Graphs Section 2.6 Page 326.
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Transcript of Rational Functions and Their Graphs Section 2.6 Page 326.
Rational Functions and Rational Functions and Their GraphsTheir Graphs
Section 2.6Section 2.6
Page 326Page 326
Rational Function- a quotient of two Rational Function- a quotient of two polynomial functions in the form polynomial functions in the form
f(x) = p(x) q(x) ≠ 0
q(x) Domain:Domain:
DefinitionsDefinitions
Example 1Example 1
Find the domain of each rational functionFind the domain of each rational function
9
3)()
9)()
3
9)()
2
2
2
x
xxhc
x
xxgb
x
xxfa
Reciprocal FunctionReciprocal Function0domain ,
1)( x
xxf
)(,0 as
)(,0 as
)(, as
0)(, as
xfx
xfx
xfx
xfx
Arrow Notation
(see page 328)
0domain ,1
)(2
xx
xf
)(,0 as
)(,0 as
)(, as
)(, as
xfx
xfx
xfx
xfxArrow Notation
Use the graph to answer the Use the graph to answer the following questions.following questions.
As x → -2As x → -2--, f(x) →, f(x) → As x → -2As x → -2++, f(x) →, f(x) → As x → 2As x → 2--, f(x) →, f(x) → As x → 2As x → 2++, f(x) →, f(x) → As x → -As x → -, f(x) → , f(x) → As x → As x → , f(x) → , f(x) →
Vertical AsymptotesVertical Asymptotes
Definition: the line Definition: the line x = ax = a is a vertical asymptote of is a vertical asymptote of the graph of a function if the graph of a function if f(x)f(x) increases or decreases increases or decreases (goes to infinity) without bound as (goes to infinity) without bound as xx approaches approaches aa
Locating Vertical Asymptotes: set the denominator of Locating Vertical Asymptotes: set the denominator of your rational function equal to zero and solve for xyour rational function equal to zero and solve for x
Find the vertical asymptotes of f(x) = Find the vertical asymptotes of f(x) = x – 1 x – 1
xx22 – 4 – 4
HomeworkHomework
Page 342 #1 - 28Page 342 #1 - 28
HolesHoles
A value where the denominator of a rational function is A value where the denominator of a rational function is equal to zero does not necessarily result in a vertical equal to zero does not necessarily result in a vertical asymptote.asymptote.
If the numerator and the denominator of the rational If the numerator and the denominator of the rational function has a common factor (x – c) then the graph will function has a common factor (x – c) then the graph will have a hole at x = chave a hole at x = c
Example: f(x) = Example: f(x) = (x(x22 – 4) – 4)
x – 2 x – 2
Finding the Horizontal AsymptoteFinding the Horizontal Asymptote
n < mn < m The horizontal asymptote is y = 0The horizontal asymptote is y = 0
n = mn = m The horizontal asymptote is the ratio The horizontal asymptote is the ratio of the leading coefficientsof the leading coefficients
n > mn > m There is no horizontal asymptoteThere is no horizontal asymptote
First identify the degree (highest power) of p(x) and q(x).
f(x) = p(x) degree nq(x) degree m
and identify their leading coefficients.
253
492
2
xx
xy
xx
xy
32
12
24
86)(
23
2
xx
xxxf
Find the Vertical and Horizontal Find the Vertical and Horizontal AsymptotesAsymptotes
Review Transformation of Review Transformation of FunctionsFunctions
Describe how the graphs of the following functions Describe how the graphs of the following functions are transformed from its parent function.are transformed from its parent function.
xxf
1)( 2
1)(
xxf
1)5(
1)(4. 6
3
1)(.3
31
)(2. 1
1)(.1
2
2
xxf
xxf
xxf
xxf
HomeworkHomework
Page 342 #29 - 48Page 342 #29 - 48
Graphing Rational FunctionsGraphing Rational Functions
Seven Step Strategy – page 334Seven Step Strategy – page 334
1.1. Check for symmetryCheck for symmetry
2.2. Find the interceptsFind the intercepts
3.3. Find the asymptotes – check for holesFind the asymptotes – check for holes
4.4. Plot additional points as necessaryPlot additional points as necessary
Example 6 – Graph Example 6 – Graph 1.1. SymmetrySymmetry
2.2. InterceptsIntercepts
3.3. AsymptotesAsymptotes
4.4. Plot pointsPlot points
4
3)(
2
2
x
xxf
Example – Graph Example – Graph 1.1. SymmetrySymmetry
2.2. InterceptsIntercepts
3.3. AsymptotesAsymptotes
4.4. Plot pointsPlot points
6
2)(
2
xx
xxf
Slant AsymptotesSlant Asymptotes Slant Asymptotes occur when the degree of the numerator of a Slant Asymptotes occur when the degree of the numerator of a
rational function is exactly one greater than that of the rational function is exactly one greater than that of the denominatordenominator
Note- when the degrees are the same or the denominator has a Note- when the degrees are the same or the denominator has a greater degree the function has a greater degree the function has a horizontalhorizontal asymptote. asymptote.
f x = x3+1
x2
6
4
2
-2
-4
-5 5
Line l is a slant asymptote for a function f(x) if the graph of y = f(x) approaches l as x → ∞ or as x → -∞
l
Determine the Slant AsymptoteDetermine the Slant Asymptote
Use synthetic division to Use synthetic division to find the slant asymptote find the slant asymptote then graph the functionthen graph the function
12
10
8
6
4
2
-2
5
2
132)(
2
x
xxxf
Find the Slant AsymptoteFind the Slant Asymptote
53
74)(
2
23
x
xxxf
use long division
Partner WorkPartner WorkCheck for symmetry then find the intercepts, Check for symmetry then find the intercepts, asymptotes, and holes of each rational functionasymptotes, and holes of each rational function
9
43)(.6
352
3)(.5
1
2)(.4
1
)1()(.3
12
3)(.2
3
1)(.1
2
23
2
2
2
x
xxxxV
xx
xxT
x
xxxP
x
xxh
x
xxg
xxf
HomeworkHomework
Page 342 #49 – 78 do 2 skip 1Page 342 #49 – 78 do 2 skip 1