Algebra 2B Notes: Chapter 8, Rational Functions
1
Chapter 8: Rational Functions
8.1 Inverse Variation
Recall from Algebra 1, that direct variation occurs when two values behave in similar ways to produce a
constant quotient value.
For example:
Jon makes $6.75 per hour. His pay each week is
Pay = 6.75 x # hours
If his # hours increases, then his pay ____________.
If his # hours decreases, then his pay ____________.
Also,
Direct Variation:
y  kx or
Pay ÷ # hours always = _______
y
k
x
where “k” is some constant of variation.
In general, for direct variation, if one quantity doubles, the other quantity also doubles. If one quantity is cut in
half, the other quantity is also cut in half. Change is directly proportional.
Inverse Variation occurs when two values behave in proportionally OPPOSITE ways to produce a constant
product value.
For example:
Susan’s mother orders two large pizzas for Susan’s birthday party. (16 slices)
# slices per person = 16 ÷ number of party guests
If Susan increases the number of party guests, then each person gets ________ pizza.
If Susan decreases the number of party guests, then each person gets ________ pizza.
Also, number of party guests x number slices per person always = ______.
Inverse Variation:
y
k
or xy  k where “k” is some constant of variation.
x
In general, for inverse variation, if one quantity doubles, the other quantity is cut in half. If one quantity is cut
down by half, the other quantity doubles. Change is inversely proportional.
Algebra 2B Notes: Chapter 8, Rational Functions
2
How to decide if a relationship between two variables is DIRECT VARIATION, INVERSE VARIATION, or
NEITHER.
How does x change?
increasing
decreasing
How does y change?
How does y change?
increasing
decreasing
increasing
decreasing
Possibly Direct
Variation
Possibly Inverse
Variation
Possibly Inverse
Variation
Possibly Direct
Variation
Check values in
Check values of
y÷x = k
xy = k
(see far
left)
k always the
same?
k always the
same?
yes
no
yes
no
direct
variation
neither
inverse
variation
neither
Example 1: Is the relationship between variables a direct variation, an inverse variation or neither? Write
equations to model the direct and inverse variations.
a)
X 1 2 4 5
Y 20 10 5 4
c)
b)
X 5 10 15 20
Y 10 20 30 40
d)
X 1 3 4 6
Y 12 4 3 2
X 0.5 1.2 2 2.5
Y 40 12 10 6
Algebra 2B Notes: Chapter 8, Rational Functions
3
Example 2:
Suppose x and y vary inversely
and x = 8 when y = 6.
y


a) Write a function that models the inverse
variation.



x











b) What does the graph of the function look like?


c) What is the value of y when x = 10?



Example 3: After a major storm, your math class volunteers to remove debris from yards. The table shows the
time t in minutes that it takes a group of n students to remove the debris from an average-sized yard.
Number of students (n) 1
3 5 14
Time in minutes (t)
225 75 45 16
a) What function models the time needed to clear
the debris from an average-sized yard relative to the
number of students who do the work?
b) How many students should there be to clear
debris from an average-sized yard in at most 25
minutes?
Example 4: Each pair of values is from an inverse variation. Find the missing value.
a) (2,5), (4,y)
b) (4,6), (x, 3)

Algebra 2B Notes: Chapter 8, Rational Functions
4
8.2 The Reciprocal Function Family
f ( x) 
Functions that model inverse variation have the form:
a
a
or y 
where x  0 .
x
x
These functions are called the “Reciprocal Function Family” and the parent function is f ( x ) 
1
.
x
Example 1:
What is the graph of y 
12
?
x

What are the x-intercepts and y-intercepts?
y




What are the asymptotes of the graph?


x













vertical asymptote:



horizontal asymptote:




What are the Domain and the Range?
Domain: all real numbers except x = _____.

Range: all real numbers except y = ______.

Algebra 2B Notes: Chapter 8, Rational Functions
y
5


Example 2:
At the right is the graph of
y
10
2
x 3



Where are the vertical and horizontal asymptotes?


x

What are the domain and range?










Domain: All real numbers except x = _____.


Range: All real numbers except y = _____.


TRANSFORMATIONS ON FUNCTIONS

You can apply transformation rules to functions in the reciprocal family just like any other function.
Example 3:
How is the given function a transformation of the parent function y 
1
?
x
What are the horizontal and vertical asymptotes?
a)
y
6
x2
b)
y
2
6
x
c)
y
2
5
3( x  1)




Algebra 2B Notes: Chapter 8, Rational Functions
6
Example 4: Sketch the asymptotes and Graph the following Rational Functions
y
a)
3
x
y
b)
2
5
x4
y
y












x














x



























Example 5: The junior class is renting a laser tag facility with a capacity of 325 people. The cost for the facility is
$1200. The party must have 13 adult chaperones.
a) If every student who attends shares the facility cost equally, what function models the cost per student, C,
with respect to the number of students, n, who attend? What is the domain of the function? How many
students must attend to make the cost no more than $7.50?
Domain = from 1 to max # of students
b) THINK ABOUT IT: What if the class wants to give away 30 free tickets to students in a drawing? Now how
many paying students must attend so the cost for each is no more than $7.50?
c) If 300 students attend, then what is the cost per student?

Algebra 2B Notes: Chapter 8, Rational Functions
7
8.3 Rational Functions and Their Graphs
A rational function is a function that you can write in the form f ( x ) 
polynomials and Q(x) ≠ 0.
P( x )
where P(x) and Q(x) are
Q( x )
The domain of the function is all real numbers EXCEPT any value of x that causes the
denominator to equal 0. Below are the graphs of three rational functions:
A
B
C
CONTINUOUS: No real value of x makes the denominator = 0. The graph has no jumps, breaks, or holes. You
can draw the graph and your pencil never leaves the paper.
Which graph(s) is continuous? _______
DISCONTINUOUS: There are values of x that cause the denominator to equal zero. The graph cannot be drawn
without your pencil leaving the paper. There may be jumps, breaks, or holes.
Which graph(s) are discontinuous? ________
y

Example 1:
Use a graphing calculator to graph.
Identify any discontinuities.


y
3
x4


x















Algebra 2B Notes: Chapter 8, Rational Functions
y
Example 2:
6 x
x3
8
y
Example 3:
8
( x  4)( x  2)
y
y










x




Example 4:





x

















y
2 x  10
x 2  25




y





x










What happens at x = -5? ______________________
What happens at x = 5 ? ______________________
Look at
y
2 x  10
x 2  25
Factor the numerator and denominator.
Can this be simplified?





Algebra 2B Notes: Chapter 8, Rational Functions
9
Sometimes you may need to factor first to see where the points of discontinuity are…
If x  a is a value of x that makes the denominator = 0, then it is a discontinuity.
If the factor in the denominator can CANCEL with a factor in numerator, then there is a point of
discontinuity or a hole in the graph at x  a .
If the factor cannot cancel, then there is a vertical asymptote at x  a .
Review Basic Factoring Patterns:
GCF:
Difference 2 Squares:
Trinomials:
x  3x
x  49
x2  5x  6
2
2
Example 5:
Find the discontinuities, if any exist, in the following functions.
State if the discontinuity is a “hole” or a “vertical asymptote.”
You may need to factor first.
a)
y
x
x2
b)
y
1
x  16
c)
y
x2 1
x2  3
d)
y
x 1
x  3x  2
e)
x 2  10 x  25
y
x3  25 x
f)
x2  4x
y 2
x  5x
2
2
3x 2  5 x  2
Algebra 2B Notes: Chapter 8, Rational Functions
10
8.4 Rational Expressions
A rational expression is the quotient of two polynomials. Think of a rational expression as a ratio or as a
fraction. A rational expression is in simplest form when its numerator and denominator are polynomials that
have no common divisors.
Simplify Rational Expression
1. If possible, factor the numerator
and denominator.
Example 1: Simplify the following rational expressions. State
any restrictions on the variable.
A.)
x 1
x 1
B)
x 2  3x  2
x3
C)
15 x
25 x 2
D)
4 x  12
x3
G)
2. Divide out common factor(s).
3. Simplify
4. Check the original expression for
restrictions on the variable. The
denominator ≠ 0.
E)
x2  x  6
x2  x  2
F)
24 x3 y 2
6 x 2 y3
H)
12  4 x
x 2  3x
I)
10  8 x
16 x 2  25
5 x 2  45
x2  5x  6
Algebra 2B Notes: Chapter 8, Rational Functions
11
Multiplying Rational Expressions
Recall:
so if
A C AC
 
B D BD
AB C

BC D
Example 2: What is the product in simplest form? State any
restrictions on the variable.
a)
2 x  8 x2  5x  4

x 2  16 x 2  8 x  16
b)
x 2  5 x  6 4 x 2  100

6 x  30 x 2  4 x  3
then
Multiplication is the ONLY TIME that you
can “cross cancel” factors!!!
1. Factor all polynomials.
2. Divide out common factors.
3. Simplify.
4. State any restrictions from the
original expression.
Dividing Rational Expressions
Recall:
A C
A D
  
B D
B C
Example 3: What is the quotient in simplest form? State any
restrictions on the variable.
a)
x2  5x  4
x2 1
 2
16 x3
2x  6x
b)
2 x
x 2  3x  10

x2  2x  1
x2 1
The first step in any division problem is to
rewrite division as multiplication by the
reciprocal. Factor all polynomials.Divide
out common factors.Simplify.State any
restrictions from the original expression.
1. Rewrite Division as
Multiplication by the reciprocal.
2. Follow same procedure as
multiplication problem.
Algebra 2B Notes: Chapter 8, Rational Functions
12
8.5 Adding and Subtracting Rational Expressions
Day 1
Add:
1 1

2 3
Common Wrong Answer
Instead: Get like denominators
2
 0.4
5
Adding or Subtracting Rational Expressions
Example 1: Simplify each sum or difference. State any
restrictions on the variable.
a)
3
6

5x 7 y
1. Factor the polynomials.
State any restrictions on the variable.
x y

B C
2. Rewrite each expression with the LCD
(Least Common Denominator)
x C
y B



B C
C B
3. Add the numerators. Combine Like
Terms
xC  yB
BC
b)
5
2

2
2 x y 5 xy 3
4. Factor the numerator and divide out
the common factors.
Algebra 2B Notes: Chapter 8, Rational Functions
13
c)
8
3

x  49 x  7
d)
2
7

x3 x5
e)
4
6x
 2
x  2 x  3x  2
f)
3
4
 2
x  25 x  6 x  5
g)
5x  4 6 x  7

3x  12 x 2  4 x
h)
y
1
 2
4y  8 y  2y
2
2
Algebra 2B Notes: Chapter 8, Rational Functions
14
Day 2: Complex Fractions
A complex fraction is a rational expression that has at least one fraction in its numerator or denominator or
both. Remember that a fraction is basically another way to express a quotient. You can use that to help you
simplify some complex fractions.
Example 2: What is a simpler form of the complex fraction?
a)
9
xy
x
5
c)
3
x 1
5
x 1
b)
x 2  81
3x
x9
8
d)
x3
x3
x2  9
3x  9
If your complex fraction has a sum or a difference in its numerator or denominator, add or subtract first (like we
did yesterday) and THEN solve like example 2 above.
Example 3.
a)
3 2

5 x
1 6

2 x
Algebra 2B Notes: Chapter 8, Rational Functions
15
8.6 Solving Rational Equations
A rational equation is an equation that contains at least one rational expression.
Example 1: What are the solutions of the rational equation?
a)
Strategy:
Solving a Rational Equation
x  30 x
1
 
12
6
2
1.
Factor the denominators
to find the LCD.
2. Multiply each side by the
LCD to clear denominators.
3. Simplify & Solve
4. CHECK YOUR ANSWER(S)
This procedure tends to
produce extraneous roots!
b)
2 x  9 19 x  7
 
7
35
5
c)
2x 1 7

5 x  2 22
5. Write the solution(s).
d)
3
7

x  5 x  15
Algebra 2B Notes: Chapter 8, Rational Functions
16
Example 2:
a)
1
3
9

 2
x  7 x  7 x  49
b)
3
5
57

 2
x  9 x  4 x  5 x  36
c)
2
1
6
 
x  3 x x( x  3)