1. Graphing rational functions
Lesson 78
Graphing rational functions

2. Rational function
A rational function is a function whose rule can be given as a rational expression.
This means that a rational expression has a variable in the denominator- they are undefined when the denominator is equal to zero.
A value of a variable for which an expression or a function is undefined is called an excluded value.

3. Find the excluded values
y= m-4
3m-12
Simplify the denominator by factoring out GCF 3(m-4)
Set the denominator equal to 0, to find the excluded value
3(m-4)= 0
m-4 = 0
m= 4 is the excluded value

4. Find the excluded values2
x-8 4
x+10

5. Discontinuous functions
Rational functions are discontinuous functions.
Discontinuous functions are functions that have a break or jump in the graph.
A break or jump in the graph can be due to an asymptote.
An asymptote is a boundary line that the graph of a function approaches but never touches or crosses

6. Vertical asymptote
To determine vertical asymptotes, find the excluded value for the function

7. Horizontal asymptotes
Rational functions also have horizontal asymptotes.
When the function is in the form y= a c
x-b
the horizontal asymptote is at y = c
When c = 0, the horizontal asymptote is the x-axis

8. Identify the asymptotes
x-8
to find vertical asymptote , find excluded value x-8 = 0,
so x = 8 is vertical asymptote
since c= 0 ,
the horizontal asymptote is y = 0

9. Identify the asymptotes
x+10
y = 5
x-1 y=
x-3

10. Graphing using asymptotes
The parent function for rational functions is
y= 1
x By identifying the asymptotes and graphing them with dotted lines, and plotting a few points, the rational function can be graphed.

11. Identify the asymptote and graph
x vertical asymptote is x = 4
horizontal asymptote is y = -2
Graph the asymptotes as dotted lines
Make a table of values (choose points on either side of the vertical asymptote) x
y /4 -5/ /2
Plot the points and connect them with smooth curves

12. Identify the asymptotes and graph
x-3
y =
x+2