Chapter 3: The Nature of Graphs
Section 3-7: Graphs of Rational Functions
Definition of a rational function
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A rational function is a quotient of two polynomial functions. It has the
form
g ( x)
f ( x) =
h( x )
where h( x) ≠ 0
The mom parent rational
function is
1
f ( x) =
x
The graph has two branches. These branches approach lines called
asymptotes. They do not cross them. There are vertical and horizontal
asymptotes. The vertical asymptote is also called the pole of the function.
Vertical & Horizontal Asymptote
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The line x=a is a vertical asymptote for a function
f (x) if f (x)
or f (x)
- ∞ as x
a
from either the left or the right.
∞
The line y=b is a horizontal asymptote for a function
f (x) if f (x)
b as x
or x
-
∞
∞
Example# 1
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Determine the asymptotes for the graph of
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First we know f(1) is undefined so there may be a
vertical asymptote at x=1. To find out, we need to
make sure that as the function f approaches infinity
that x approaches 1. If we set up at table, we can see
that this is true. As x approaches 1, f (x) approaches
negative infinity. Thus we have a vertical asymptote
at x=1
x
f ( x) =
x −1
x
f (x)
.9
-9
.99
-99
.999
-999
.9999 -9999
Two methods to check for a horizontal
asymptote
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Method 1:
Take the function and solve for x in terms of y.
x
y=
→ y( x −1) = x → xy − y = x
x −1
→ xy − x = y → x( y −1) = y →
y
x=
y −1
The rational expression is
undefined for y=1. Thus
the horizontal asymptote is
the line y=1
Method 2
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First divide the numerator and the denominator by
the highest power of x.
x
y=
x −1
y=
x
x
x 1
−
x x
As the value of x increases,
1
the value of
approach 0.
x
Therefore the value of the
entire expression
approaches 1
Parent Graph
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As we did with the other families of graphs, we can use the
mom and the dad parent graph to graph rational function
children.
The pattern of transformations remains the same no matter
which family you are graphing.
So to describe the transformations that take place in the
function
6
m (x) = −
„
x+2
− 4
We know it is from the dad parent (which is a reflection of the
mom over the x axis). It stretches vertically by a factor of 6
and then translates 2 units to the left and 4 units down. The
new vertical asymptote is x=-2 and the horizontal asymptote
changes from y=0 to y=-4
Slant asymptote
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There exist a third type of asymptote called a slant
asymptote. Slant asymptotes occur when the degree of the
numerator of a rational function is exactly one greater than
that of the numerator.
When the degrees are the same or the denominator has a
greater degree, the function has a horizontal asymptote.
Definition: A line M is a slant asymptote of a function f (x) if
the graph y=f (x) approaches the line M as x approaches
infinity.
Example
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Determine the slant asymptote for
First use division to rewrite the function
4 x − 10 −
„
3
x+4
x + 4 4 x 2 + 6 x − 37
− (4 x 2 + 16 x)
_____________
− 10 x − 37
− (−10 x − 40)
_________
3
f ( x) =
4 x 2 + 6 x − 37
x+4
As x approaches infinity, 3/x+4
approaches 0. So the graph of f (x)
will approach y=4x-10 which is the
slant asymptote for the graph. Note
that x=4 is a vertical asymptote.
This method
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This method of using division can also be
used to find a horizontal asymptote when the
degree of the numerator is equal to or
greater than that of the denominator.
HW#21
Section 3-7
„ Pp. 186-188
„ #15,16,17,18,20,21,22,25,26,30,31,49,56
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