### Graphing Rational Functions

```ADV122
GRAPHING RATIONAL FUNCTIONS
Warm Up
Graph the function
=− +3
2
+4
GRAPHING RATIONAL FUNCTIONS
We have graphed several functions,
now we are adding one more to the
list!
Graphing Rational Functions
GRAPHING RATIONAL FUNCTIONS
Parent Function:   =

GRAPHING RATIONAL FUNCTIONS
Pay attention to the transformation clues!
(-a indicates a reflection
in the x-axis)
a
f(x) =
+k
x–h
vertical translation
(-k = down, +k = up)
horizontal translation
(+h = left, -h = right)
Watch the negative sign!! If
h = -2 it will appear as x + 2.
GRAPHING RATIONAL FUNCTIONS
Asymptotes

Places on the graph the function will approach,
but will never touch.
GRAPHING RATIONAL FUNCTIONS
1
Graph: f(x) =
x
Vertical Asymptote: x = 0
Horizontal Asymptote: y = 0
No horizontal shift.
No vertical shift.
A HYPERBOLA!!
GRAPHING RATIONAL FUNCTIONS
W    =

−

look like?
GRAPHING RATIONAL FUNCTIONS
1
Graph: f(x) =
x+4
x + 4 indicates a
shift 4 units left
Vertical Asymptote: x = -4
No vertical shift
Horizontal Asymptote: y = 0
GRAPHING RATIONAL FUNCTIONS
1
Graph: f(x) =
–3
x+4
x + 4 indicates a
shift 4 units left
Vertical Asymptote: x = -4
–3 indicates a shift 3
units down which
becomes the new
horizontal asymptote
y = -3.
Horizontal Asymptote: y = 0
GRAPHING RATIONAL FUNCTIONS
Graph: f(x) =
x
+6
x–1
x – 1 indicates a
shift 1 unit right
Vertical Asymptote: x = 1
+6 indicates a shift 6
units up moving the
horizontal asymptote
to y = 6
Horizontal Asymptote: y = 1
GRAPHING RATIONAL FUNCTIONS
You try!!
1
1.  =
+2

2.  =
1
+3
−4
GRAPHING RATIONAL FUNCTIONS
How do we find asymptotes
based on an equation only?
GRAPHING RATIONAL FUNCTIONS
Vertical Asymptotes (easy one)

Set the denominator equal to zero and solve for
x.

Example:  =
6
−3

x-3=0
x=3

So: 3 is a vertical asymptote.
GRAPHING RATIONAL FUNCTIONS
Horizontal Asymptotes (H.A)


In order to have a horizontal asymptote, the
degree of the denominator must be the same, or
greater than the degree in the numerator.
Examples:
2 −3
  =
+7
3 −2
  = 3
−2
+1
  = 2

No H.A because 2 > 1
Has a H.A because 3=3.
Has a H.A because 1 < 2
GRAPHING RATIONAL FUNCTIONS
3 cases
GRAPHING RATIONAL FUNCTIONS
If the degree of the denominator
is GREATER than the
numerator.

The Asymptote is y=0 ( the x-axis)
GRAPHING RATIONAL FUNCTIONS
If the degree of the denominator
and numerator are the same:

Divide the leading coefficient of the numerator
by the leading coefficient of the denominator in
order to find the horizontal asymptote.
6 3
3 3 −2

Example:  =

Asymptote is 6/3 =2.
GRAPHING RATIONAL FUNCTIONS
If there is a Vertical Shift


The asymptote will be the same number as the
vertical shift.
(think about why this is based on the examples
we did with graphs)
5
+7
−3

Example:

Vertical shift is 7, so H.A is at 7.