RATIONAL

1
RATIONAL
FUNCTIONS II
GRAPHING RATIONAL FUNCTIONS
2
Steps to GraphingRational Functions
3
(No Transcript)
4
So lets plot the y intercept which is (0, - 1)
5
If the numerator of a fraction 0 then the whole
fraction 0 since 0 over anything 0
But 0 6 is not true which means there IS NO x
intercept.
6
Not the original and not negative of function so
neither even nor odd.
7
degree of the top 0
remember x0 1
degree of the bottom 2
If the degree of the top is less than the degree
of the bottom the x axis is a horizontal
asymptote.
8
Choose an x on the right side of the vertical
asymptote.
Choose an x on the left side of the vertical
asymptote.
x
R(x)
-4
0.4
1
-1
4
1
Choose an x in between the vertical asymptotes.
9
Pass through the point and head towards asymptotes
Pass through the point and head towards asymptotes
There should be a piece of the graph on each side
of the vertical asymptotes.
Go to a function grapher or your graphing
calculator and see how we did.
Pass through the points and head towards
asymptotes. Cant go up or it would cross the x
axis and there are no x intercepts there.
10
The window on the calculator was set from -8 to 8
on both x and y.
Notice the calculator draws in part of the
asymptotes, but these ARE NOT part of the graph.
Remember they are sketching aids---the lines that
the graph heads towards.
11
Let's try another with a bit of a "twist"
vertical asymptote from this factor only since
other factor cancelled.
But notice that the top of the fraction will
factor and the fraction can then be reduced.
We will not then have a vertical asymptote at x
-3, but it is still an excluded value NOT in the
domain.
12
We'll graph the reduced fraction but we must keep
in mind that x ? - 3
So lets plot the y intercept which is (0, - 1/3)
13
If the numerator of a fraction 0 then the whole
fraction 0 since 0 over anything 0
x 1 0 when x -1 so there is an x intercept
at the point (-1, 0)
14
Not the original and not negative of function so
neither even nor odd.
15
degree of the top 1
1
1
degree of the bottom 1
If the degree of the top equals the degree of the
bottom then there is a horizontal asymptote at y
leading coefficient of top over leading
coefficient of bottom.
16
We already have some points on the left side of
the vertical asymptote so we can see where the
function goes there
x
S(x)
4
5
6
2.3
Let's choose a couple of x's on the right side of
the vertical asymptote.
17
Pass through the point and head towards asymptotes
Pass through the points and head towards
asymptotes
There should be a piece of the graph on each side
of the vertical asymptote.
REMEMBER that x ? -3 so find the point on the
graph where x is -3 and make a "hole" there since
it is an excluded value.
Go to a function grapher or your graphing
calculator and see how we did.
18
The window on the calculator was set from -8 to 8
on both x and y.
Notice the calculator drew the vertical asymptote
but it did NOT show the "hole" in the graph. It
did not draw the horizontal asymptote but you can
see where it would be at y 1.