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