2.6 & 2.7 Rational Functions and Their Graphs 2.6 & 2.7 Rational Functions and Their Graphs...
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Transcript of 2.6 & 2.7 Rational Functions and Their Graphs 2.6 & 2.7 Rational Functions and Their Graphs...
![Page 1: 2.6 & 2.7 Rational Functions and Their Graphs 2.6 & 2.7 Rational Functions and Their Graphs Objectives: Identify and evaluate rational functions Graph.](https://reader036.fdocuments.us/reader036/viewer/2022062517/56649eb75503460f94bc09a6/html5/thumbnails/1.jpg)
2.6 & 2.7 Rational Functions 2.6 & 2.7 Rational Functions and Their Graphsand Their Graphs
2.6 & 2.7 Rational Functions 2.6 & 2.7 Rational Functions and Their Graphsand Their Graphs
Objectives: •Identify and evaluate rational functions•Graph a rational function, find its domain and range, write equations for its asymptotes, identify any holes in its graph, and identify the x- and y- intercepts
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What is a Rational Expression?
• A rational expression is the quotient of two polynomials.
• A rational function is a function defined by a rational expression.
2
3
5
3
( )( 4)( 4)
( 3)( )
27
xy
xx
g xx x
xf x
x
3
1
2
7
( )2 1
( )5
x
x
yx
xg x
x
xf x
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Simplify
2
2
x 7x 18x 8x 9
2
2
x 7x 18x 8x 9
(x 9)(x 2)(x 9)(x 1)
x 2x 1
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Find the Domain
Find the domain of 2
2
4 21( )
9 36
x xh x
x x
To find the domain of a rational function, you 1st must find the values of x for which the denominator equals 0. x2 – 9x – 36 = 0
(x – 12)(x + 3) = 0 x = 12 or -
3
The domain is all real numbers except 12 and -3.
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Vertical Asymptotepronounced… “as-im-toht”
In a rational function R, if (x – a) is a factor of the denominator but not a factor of the numerator, x = a is vertical asymptote of the graph of R.
What is an asymptote?
•It is a line that a curve approaches but does not reach.
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To find vertical asymptotes
1. Find the zeros of the denominator2. Factor numerator3. Simplify fraction4. There are vertical asymptotes at
any factors that are left in the denominator
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Identify all vertical asymptotes of
2
3( )
3 2
xr x
x x
Step 1: Factor the denominator.
Step 2: Solve the denominator for x.
Equations for the vertical asymptotes are x = 2 and x = 1.
3( )
( 2)( 1)
xr x
x x
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More Practice
2
2
6( )
9
x xf x
x
( 2)( 3)
( 3)( 3)
x x
x x
Identify the domain and any vertical
asymptotes. 2
2
6( )
9
x xf x
x
D: All Real #’s except x=-3,3
VA: at x=-3
2
3
x
x
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Look at the table for this function:
2
2
6 ( 2)( 3) 2( )
9 ( 3)( 3) 3
x x x x xf x
x x x x
We can understand why the -3 shows an “error” message.
Buy why does the 3 also show an “error” message?
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That means there is a “Hole” in the graph…
2
2
6 ( 2)( 3) 2( )
9 ( 3)( 3) 3
x x x x xf x
x x x x
That is what happens to the part we “cross
off” the fraction. That is where the hole(s) is.
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Holes in GraphsIn a rational function R, if x – b is a factor of the numerator and the denominator, there is a hole in the graph of R when x = b (unless x = b is a vertical asymptote).
There is a vertical asymptote at x=-3.
And a hole at x=3.
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Horizontal Asymptote
•If degree of P < degree of Q, thenthe horizontal asymptote of R is y = 0.
R(x) = is a rational function;
P and Q are polynomials
P
Q
2( )
2 3
x smallf x
x x bigger
So… HA: y=0
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Horizontal Asymptote
R(x) = is a rational function;
P and Q are polynomials
P
Q
•If degree of P = degree of Q and a and b are the leading coefficients of P and Q, then
the horizontal asymptote of R is y = .
a
b2
2
16( )
4 5
x samef x
x x same
So… HA: y = 1
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Horizontal Asymptote
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Horizontal Asymptote
R(x) = is a rational function;
P and Q are polynomials
P
Q
•If degree of P > degree of Q, thenthere is no horizontal asymptote
3
2
7( )
4 3
x biggerf x
x x small
So… HA: D.N.E.
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Horizontal Asymptotes
0
. . .
smallHA is y
bigger
same aHA is y
same bbigger
HA D N Esmall
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Slant Asymptote
2 3 24 3 0 0 7x x x x x x
3 24 3x x x
A Slant Asymptote occurs when the degree of the numerator is exactly one degree higher than the degree of the denominator.3
2
7( )
4 3
x biggerf x
x x small
HA: D.N.E.
( ) 24 3x x
74
24 16 12x x ( ) 13 12x
Therefore:
Slant asymptote is
Y =
4x
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Let . Identify the domain
and range of the function, all asymptotes and all
intercepts. Oh, also are there any holes?
3
2( )
20
xR x
x x
3
( )( 5)( 4)
xR x
x x
Equations for the vertical asymptotes are x = -5 and x = 4.
Because the degree of the numerator is greater than the degree of the denominator, the graph has no horizontal asymptotes, but slant asymptote is y = x - 1.
D: x ‡ -5, 4
R: ???
ONLY Intercept is ( 0 , 0 )
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1Let . Find Domain & Range. Identify all asymptotes, holes and all Intercepts.
2
2
2 1( )
9
xR x
x
22 1( )
( 3)( 3)
xR x
x x
Vertical asymptotes: x = -3 and x = 3, but NO holesHorizontal asymptotes: 2
1leading coefficients
numerator and denominator have the same degree
y = 2
D: x ‡ 3, -3
R: y ‡
2
x -intercept: ( ½ √2, 0 ) ( -½ √2, 0 )
Y – intercept: ( 0 , 1/9 )
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Identify all Critical Values in the graph of the rational function, then graph.
f(x) = 2x2 + 2x
x2 – 1factor: f(x) =
2x(x + 1)
(x + 1)(x – 1)
hole in the graph:x = –1
vertical asymptote:x = 1
horizontal asymptote:y = 2
D: x ‡ 1, -1
R: y ‡ 2
Intercepts:
( 0, 0 ) ( -1, 0)
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To graph rational functions
1. Simplify function any restrictions should be listed.
2. Plot y intercept (if any)3. Plot x intercepts ( zeros of the top)4. Sketch all asymptotes (dash lines)5. Plot at least one point between each x intercept
and vertical asymptote6. Use smooth curves to complete graph
7.
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For , identify all
Critical Values, then graph the function.
2 2 4( )
2 1
x xg x
x
D: Holes:
V.A.:
H.A.: R:
S.A.:
X-intercepts:
Y-intercepts:
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For , identify all
Critical Values, then graph the function.
2
2
25( )
2 7 15
xg x
x x
D: Holes:
V.A.:
H.A.: R:
S.A.:
X-intercepts:
Y-intercepts:
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-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
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homeworkp. 152 7-12, 13-18p. 161 9, 15,23,56,61