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Characteristics of Rational Functions
Dr. Shildneck
Fall, 2014
Asymptotes
1. How do you find the vertical asymptote?
Set the denominator equal to zero
and solve for x.
Asymptotes
2. How do you find the horizontal asymptote?
a) If in the form it is just y = k.
b) If in the form it is y = a/c.
( )a
f x kx h
( )ax b
f xcx d
Transformations
The transformations of functions that you should look for are:
a) Vertical Shift
b) Horizontal Shift
c) Stretch or Shrink
d) Reflection
Intercepts
The intercepts of the function are where the graph crosses the x or y axes.
a) What are the coordinates of a point on the x-axis?
b) What are the coordinates of a point on the y-axis
Intercepts
So how can I use those facts to find
a) The x-intercept?
b) The y-intercept?
Domain and Range
For Basic Functions, Domain and Range are simple. a) Domain: All real numbers except for the vertical
asymptote(s).
b) Range: All real numbers except for the horizontal asymptote.
(this changes for more complex functions)
Intervals of Increase/Decrease
Again, for Basic Functions, these can be simple. Since there are two branches, you just have to look at the graph from left to right. There will be two intervals.
The intervals will be based on the infinities and the vertical asymptote.
(For more complex functions, this is more complicated)
End Behavior
Once more, for Basic Functions, these are simple.
Since there are two branches, you just have to look at the graph from left to right. There will be two behaviors, as you move left and right out from the center.
End Behavior
Moving right from the vertical asymptote, what y-value does the right branch approach?
Moving left from the vertical asymptote, what y-value does the left branch approach?
End Behavior Notation
Right:
As
Left:
As
,x
,x
( ) #f x
( ) #f x
Example •
3( ) 4
7f x
x
Example •
6 1( )
2 4
xf x
x
ASSIGNMENT
ASSIGNMENT 2
Characteristics of Rational Functions
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