ALGEBRA II HONORS/GIFTED
@
SECTION 8-3 – RATIONAL
FUNCTIONS and THEIR
GRAPHS
AN ALGEBRA FABLE : Ryan the Bug is crawling along
the graph f(x) = 2x + 2
. He starts on the graph at
x -x-6
x = -4 . Ryan is hungry, so he heads to the Origin Café.
Suddenly at x = -2, Ryan falls off the face of the Earth.
What happened?
Cyndi the worm is crawling along the same graph,
beginning at x = 5. She wants to have a rockin’ time.
So, Cyndi goes to (where else?) the Origin Café.
However, Cyndi’s trip is cut short when she runs into
a wall at x = 3. What happened?
x +2
on your
. Hit <zoom6>
2
x -x-6
to graph. Note there is a vertical line at x = 3. What is
the name of that vertical line?
1) Graph f(x) =
ANSWER : asymptote
Since the slopes of vertical lines are undefined or go
to infinity, the branches of the graph nearest the
asymptote go to infinity.
Asymptotes are like
you can’t touch.
. You can go close, but
2) Now, hit <zoom4> to graph. What do you notice?
ANSWER : Yes, there is a
hole in your graph! Note
it is at x = -2.
What does a hole (removable discontinuity) mean?
13x = 13
13x = ?
So,
in
every
case
. Let’s see.
x
x
7x = ?
x
10x = ?
x
So, 0 = any number.
0
100x = ?
x
It is indeterminate.
3) Graph f(x) = 2x + 2
x -x-6
x +2
f(x) = 2x + 2 =
x - x - 6 (x - 3)(x + 2)
= 1
x -3
Factor
Cancel x + 2
Note : at x=-2, f(x) is indeterminate. The “hole”
comes from the factor that cancels.
Substituting -2 into f(x) = 1 , we get -1 for f(x).
5
x -3

So, the hole is located at  -2,

-1 
5 
The asymptote is at x = 3. Note, when x = 3,
x – 3 = 0.
DISCONTINUITY
ASYMPTOTE
(factor does not
cancel)
REMOVABLE (HOLE)
(“the hole cancels”)
Remember : discontinuities make
the denominator equal to 0.
f(x) = 1
x -3
x
(hole) -2
hole :  -2, -1 

y
-1/5
5 
asymptote : x = 3
2
x
4) Graph f(x) = 2 - 3x
x - 5x + 6
x
y
5) Graph f(x) =
x
1
x2 - 3x - 10
y
For more information, access :
http://www.analyzemath.com/RationalGraphTest
/RationalGraphTutorial.html
http://www.analyzemath.com/Graphing/GraphR
ationalFunction.html
HORIZONTAL ASYMPTOTES
Given the polynomial f(x) = p(x)
q(x)
1) If the degree of p(x) is less than the degree of q(x), there is a
horizontal asymptote at y = 0. For example, f(x) = 2 x
has
x +2
a horizontal asymptote at y = 0.
2) If the degree of p(x) is equal to the degree of q(x), there is a
horizontal asymptote at y = a where a is the lead coefficient of
b
p(x) and b is the lead coefficient of q(x). For example, for
f(x) =
x2
3x2 + 2
the horizontal asymptote is
y=1 .
3
3) If the degree of p(x) is greater than the degree of q(x), there is
4
no horizontal asymptote. For example, f(x) = x
has no
x +1
horizontal asymptote.