Transcript 2 6 Rational Functions II

Objectives:
1. To find the slant
asymptotes of a
rational function
2. To graph rational
functions
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Assignment:
P. 194: 27-34 (Some)
P. 194: 35-46 (Some)
P. 194: 51-58 (Some)
P. 194-5: 59-64
(Some)
HW Supplement I
(Some)
HW Supplement II
(Some)
Asymptote
Horizontal Asymptote
Vertical Asymptote
Slant Asymptote
You will be able
asymptote of a
to find the slant
rational function
Consider the rational function below.
2 x2
f ( x)  2
x 1
2
2 2
x 1
x2  1 2 x2  0 x  0
− 2x 2
− 2
2
We know that since d = n, f
has a horizontal asymptote
at y = 2.
Since a rational function is
telling us to divide, let’s do
so.
Consider the rational function below.
2 x2
f ( x)  2
x 1
We know that since d = n, f
has a horizontal asymptote
at y = 2.
So f (x) can be rewritten as:
2
f ( x)  2  2
x 1
Approaches 0 as x → ∞
And our graph is trying to look like y = 2 at large values of x.
Consider the rational function below.
We know that since d < n, f
has no horizontal
asymptotes.
2 x Since a rational function is
2x  2
telling us to divide, let’s do
x 1
2
3
2
x  1 2 x3  0 x  0 x  0
so.
2 x3
f ( x)  2
x 1
− 2x
− 2x
2x
Consider the rational function below.
2 x3
f ( x)  2
x 1
We know that since d < n, f
has no horizontal
asymptotes.
So f (x) can be rewritten as:
2x
f ( x)  2 x  2
x 1
Approaches 0 as x → ∞
And our graph is trying to look like y = 2x at large values of x.
This is called a slant asymptote.
A rational function has a slant asymptote if
n=d+1
– The degree of the numerator is one more than the
degree of the denominator
To find the equation of a slant asymptote, use long
division and forget about the remainder.
−
At large values of x, the remainder approaches 0
anyway.
Can a rational function have both a slant
asymptote and a horizontal asymptote?
Find all asymptotes of the rational function.
2 x 2  15 x  8
f ( x) 
x3
You will be able to graph
rational functions
To graph a rational function:
1. Factor N(x) and D(x).
2. Find vertical asymptotes (where D(x) = 0) and plot as
dashed lines.
–
If a factor cancels, it is not an asymptote (A Hole)
3. Find horizontal asymptote (comparing d and n) and plot as
a dashed line.
4. Find slant asymptote (by long division w/o the remainder)
and plot as a dashed line.
5. Plot x- and y-intercepts.
–
If a factor cancels, it is not a zero (A Hole)
6. Use smooth curves to finish the graph.
Vertical Asymptotes:
• Your graph can never cross one!
• If x = a is a vertical asymptote, then interesting
things happen really close to a:
– f (x) could approach +∞ or −∞
– Think of vertical asymptotes as black holes that
attract values near a
Vertical Asymptotes:
The end behavior around a vertical asymptote is similar to
that of polynomials:
V.A. at x = 1 (multiplicity of 1)
V.A. at x = 1 (multiplicity of 1)
Vertical Asymptotes:
The end behavior around a vertical asymptote is similar to
that of polynomials:
V.A. at x = 1 (multiplicity of 2)
V.A. at x = 1 (multiplicity of 2)
Horizontal Asymptotes:
• Your graph can cross
one!
• Attracts values
approaching +∞
or −∞
Slant Asymptotes:
• Your graph can cross
one of these, too!
• Attracts values
approaching +∞
or −∞
Graph:
f ( x) 
1
x3
Graph:
f ( x) 
2x 1
x
Graph:
f ( x) 
3x
x2  x  2
Graph:
f ( x) 
3x
( x  2)( x  1)2
Graph:
f ( x) 
3x
( x  2)2 ( x  1) 2
Graph:
x2  4
f ( x)  2
x  4x  4
Graph:
x2  x
f ( x) 
x 1
Graph:
1  2x2
f ( x) 
x
Graph:
x2  x  2
f ( x)  3
x  2 x2  5x  6
Graph:
x3
f ( x)  2
2x  8
Objectives:
1. To find the slant
asymptotes of a
rational function
2. To graph rational
functions
•
•
•
•
•
•
Assignment:
P. 194: 27-34 (Some)
P. 194: 35-46 (Some)
P. 194: 51-58 (Some)
P. 194-5: 59-64
(Some)
HW Supplement I
(Some)
HW Supplement II
(Some)