Consider the rational function below.
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Transcript of Consider the rational function below.
Consider the rational function below. 2
2
2( )1
xf xx
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.
2 21 2 0 0x x x 2
22x 2−−2
2
21x
Consider the rational function below. 2
2
2( )1
xf xx
We know that since d = n, f has a horizontal asymptote at y = 2.
2
2( ) 21
f xx
So f (x) can be rewritten as:
Approaches 0 as x → ∞
And our graph is trying to look like y = 2 at large values of x.
Consider the rational function below. 3
2
2( )1
xf xx
We know that since d < n, f has no horizontal asymptotes.
Since a rational function is telling us to divide, let’s do so.
2 3 21 2 0 0 0x x x x 2x
32x 2x−−2x
2
21
xx
Consider the rational function below. We know that since d < n,
f has no horizontal asymptotes.
2
2( ) 21
xf x xx
So f (x) can be rewritten as:
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.
3
2
2( )1
xf xx
Slant AsymptotesA 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.
Exercise 1Find all asymptotes of the rational function.
22 15 8( )3
x xf xx
To Graph: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.
More on AsymptotesVertical 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
More on AsymptotesVertical 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)
More on AsymptotesVertical 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)
More on AsymptotesHorizontal
Asymptotes:• Your graph can
cross one!• Attracts values
approaching +∞ or −∞
More on AsymptotesSlant Asymptotes:• Your graph can
cross one of these, too!
• Attracts values approaching +∞ or −∞
Exercise 2Graph:
1( )3
f xx
Exercise 3Graph:
2 1( ) xf xx
Exercise 4Graph:
2
3( )2
xf xx x
Exercise 5Graph:
2
3( )( 2)( 1)
xf xx x
Exercise 6Graph:
2 2
3( )( 2) ( 1)
xf xx x
Exercise 7Graph:
2
2
4( )4 4
xf xx x
Exercise 8Graph:
2
( )1
x xf xx