Name: __________________________________________________________ Date: _______________ Block: _____
Chapter 9: Rational Expressions
Simplifying and Multiplying Rational Expressions
Simplify the following, state and restrictions:
12 x 2 y 3
1.
18 x5 y
4.
x2  5x  6
x5  36 x3
2.
2  x  5
3 x  5
5.
x2  x  6
3x  3
 2
2
x  2 x  3 4 x  8 x  12
3.
4  x  4  2 x  1
18  2 x  1 x  2 
Dividing Rational Expressions
Divide and simplify completely:
1.
x4
 x  2 2

x 2  3x  4
x2  4
2.
x 2  3x  4 x 2  4 x

x2  2x  3 x2  4
Add and Subtract Rational Expressions with Like Denominators
Add/Subtract
Fractions and
Rational
Expressions
1.
2.
3.
4.
Make sure the denominators are the same.
Add or subtract the numerators (combine like terms) and keep the denominators the same.
Factor the numerator and/or denominator, if possible
Simplify, if possible
Simplify the following completely:
1.
4
6

x2 x2
2.
25 x 2
9

5x  3 5x  3
3.
2 x 2  7 x  15 x 2  4 x  19

2 x2  7 x  4 2 x2  7 x  4
Name: __________________________________________________________ Date: _______________ Block: _____
Add and Subtract Rational Expressions with Unlike Denominators
Add/Subtract
UNLIKE Rational
Expressions
1. Determine the least common multiple to get a common denominator! You might need to
factor each denominator first…
2. Figure out which factor is missing and multiply the numerator and denominator by the
missing factor(s).
3. Simplify the numerator (FOIL, distribute, combine like terms, etc).
4. Factor the numerator, if possible.
5. Simplify, if possible.
Simplify the following rational expressions.
1.
3x  2 4 x  5

4
8
2.
7
1

2
3x 9 x
3.
4.
6x
4x

3x  1 2 x  5
5.
x
5
 2
x  3 x  6x  9
6.
3
5

x2 x4
x 1
3
x2

 2
x2 x2 x 4
Solving Rational Equations
Solving a
Rational
Equation
•
•
•
•
Find the least common denominator (LCD)
FRACTION BUSTERS: multiply every term by the LCD to eliminate the denominators
Solve the new equation and check to make sure your solution works
Be sure to watch for domain restrictions.
– Values of x that are not possible
Solve:
1.
x
1

x6 x4
2.
x
2x
18

 2
x 3 x 3 x 9
3.
3
12
1
 2

x2 x 4 x2
4.
x 1 2
x
 
x 1 x x 1
Name: __________________________________________________________ Date: _______________ Block: _____
Restrictions of Rational Functions
Finding
Restrictions
•
Remember that domain tells you what x-values can be substituted into the equation.
•
For example, for f ( x ) 
•
don't want the denominator to become zero! So the restriction is: x ≠ 0
In general, to find the restrictions of a rational function, figure out what number(s) would
cause the denominator to become zero.
1
you can substitute in any number in for x except x = 0 because we
x
Find the restrictions of each function:
1.
f  x 
1
x4
2. f  x  
 3x  1 x  2 
 x  3 4 x  7 
3. f  x  
x 2  7 x  12
x 2  9 x  20
HORIZONTAL ASYMPTOTES
Horizontal Asymptotes
• (HA) y=
• A Horizontal Asymptotes tells the end behavior
of a function. To determine the y-value of the
HA, we need to compare the powers of the
numerator and denominator.
•
•
•
If the powers are the same, divide the coefficients.
If the powers are larger in the denominator, the HA
is y = 0.
If the powers are larger in the numerator, there is
no HA.
Determine the y value of the horizontal asymptote:
2 x2  3
3x 2  4 x  5
2. g  x  
3x 2  3
4 x3  1
 x  1 2 x  3
 x  1 x  4 
4. f  x  
3x3  4 x  2
x 1
1. f  x  
3. y 
VERTICAL ASYMPTOTES AND HOLES
Vertical Asymptotes


Holes
(VA)
x=
•
If a factor CANNOT be canceled and remains in the denominator, it creates a
vertical asymptote.
•
If a factor CAN be canceled from the numerator and denominator, then it creates a
hole.
Name: __________________________________________________________ Date: _______________ Block: _____
• REMEMBER—YOU MUST FACTOR COMPLETELY FIRST!
Determine the x-values for any holes or asymptotes:
1. f  x  
4. y 
x3
 x  3 x  2 
3x  1
3x 2  x
2. g  x  
x 1
 x  2 x  1
3. h  x  
5. f  x  
x2  6 x  8
x 2  16
6. y 
x
x  x  1
x 2  10 x  21
x7
Translations
f ( x) 
a
k
xh
Translations:



h ~ the horizonal shift
k ~ the vertical shift
a ~ the stretch ( a >1), shrink (0 < a < 1), or reflection (a < 1)
Describe the translations of the following functions.
1. f ( x )  
3
4
x2
2. f ( x) 
1
1
x5
horizontal shift:______________
horizontal shift:______________
vertical shift:______________
vertical shift:______________
stretch, shrink, and/or reflection:_______________
stretch, shrink, and/or reflection:_______________
3. f ( x )  
1
3
x
4. f ( x) 
1
x6
horizontal shift:______________
horizontal shift:______________
vertical shift:______________
vertical shift:______________
stretch, shrink, and/or reflection:_______________
stretch, shrink, and/or reflection:_______________