Prep for College Algebra 6.1B Notes
Graphing Rational Functions
Name________________________
𝒑(𝒙)
A rational function is any function that can be written as𝒇 𝒙 = 𝒒(𝒙) , where 𝒑 𝒙 and 𝒒(𝒙) are
both functions and 𝒒(𝒙) ≠ 𝟎.
Asymptote - A line that the graph never crosses.
1. Graph the following rational functions. Describe for what values of x the functions are
undefined. Identify the vertical asymptotes of each function.
!
!
a) 𝑦 = !!!
b) 𝑟 𝑥 = !!! !!!
Vertical asymptote(s)
__________________
c) 𝑔 𝑥 =
!!!!
! ! !!
Vertical asymptote(s)
__________________
Vertical asymptote(s)
______________________
!!!
d) ℎ 𝑥 = ! ! !!!!!
Vertical asymptote(s)
__________________
What do you notice about the degree of the numerator and the denominator in each function?
Where are the horizontal asymptotes? _______________
2. In each of the following rational functions: Identify the degree of the numerator and the
degree of the denominator. Graph the function. Identify all vertical and horizontal
asymptotes.
a) 𝑓 𝑥 = !!!!
!!!
! ! !!!!!
!!!!
b) 𝑔 𝑥 = !!!!
c) ℎ 𝑥 = ! ! !!!!!
Degree (numerator) = _____
Degree (numerator) = _____
Degree (numerator) = _____
Degree (denominator) = _____
Degree (denominator) = _____
Degree (denominator) = _____
HA: _______ VA: _______
HA: _______ VA: _______
HA: _______ VA: _______
What do you notice about the degree of the numerator and the denominator in each function?
Where are the horizontal asymptotes? _______________
Write a rule for finding the horizontal asymptote based on these examples.
Holes (points of discontinuity) in Rational Functions.
A hole will occur in the graph of a rational function when you have the same factor in the numerator
and the denominator.
Factor the numerator and denominator of this function.
𝑓 𝑥 =
𝑥! − 4
𝑥 ! − 7𝑥 + 10
Reduce the equation and graph the reduced equation.
Find the coordinate of the hole or point of discontinuity in the graph. Graph the hole.
Graph each rational function by first finding the asymptotes and/or holes and the x and yintercepts of the graph.
1. 𝑓 𝑥 =
!!! !!!
!!! !!!
!!!!"
2. 𝑔 𝑥 = ! ! !!!!!"
Prep for College Algebra 6.1B
Homework – Graphing Rational Functions.
Name___________________
Find the asymptotes and/or holes, and x- and y-intercepts of each rational
function.
1. 𝑔 𝑥 = !!!
!!! !!!
2. 𝑓 𝑥 =
! ! !!!!!
! ! !!
!
3. 𝑦 = ! ! !!!
Find the vertical and horizontal asymptotes, x- and y-intercepts, and/or holes for the graph of each rational
function. Then graph the function. Verify your graph by graphing on your calculator.
x
5− x
4. y = 2
5. y =
x ( x − 1)
x −1
VA: _________ HA: __________
VA: _________ HA: __________
Hole:___________
Hole:___________
6.
𝑓 𝑥 = !!!
! ! !!"
7. y =
x+2
x−4
VA: _________ HA: __________
VA: _________ HA: __________
Hole:___________
Hole:___________
!! ! !!
3
x−2
8. 𝑦 = ! ! !!"
9. y =
VA: _________ HA: __________
VA: _________ HA: __________
Hole:___________
Hole:___________
10. y =
( x − 2)(2 x + 3)
(5 x + 4)( x − 3)
11. 𝒇 𝒙 =
𝒙𝟐 !𝟏𝟎𝒙!𝟐𝟏
𝒙𝟐 !𝟐𝒙!𝟑
VA: _________ HA: __________
VA: _________ HA: __________
Hole:___________
Hole:___________