Name____________________________
Hour__________
Unit 6 Notes
Rational Functions
1
Algebra 2 Unit 6
Learning Goal: Students will be able to multiply, divide, add and subtract rational expressions, as well as
solve rational equations and graph rational functions.
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Objective in words
Students will be able to simplify rational expressions.
Students will be able to multiply rational expressions.
Students will be able to divide rational expressions.
Students will be able to add rational expressions, finding the LCD if necessary.
Students will be able to subtract rational expressions, finding the LCD if necessary.
Students will be able to solve rational equations by cross multiplication or multiplying
by the LCD.
7
a
 k using the
Students will be able to graph rational functions of the form f ( x) 
xh
asymptotes.
8
p ( x)
Students will be able to graph rational functions of the form f ( x) 
using the
q ( x)
asymptotes and x-intercepts.
Score 4
I can perform all operations with rational expressions, even when there are multiple steps
involved including factoring and finding an LCD. I can also solve rational equations of all
types with ease and graph rational functions of any type, understanding where the
asymptotes are.
Score 3
I can perform all operations with rational expressions as well as solve rational equations
and graph rational functions with little error.
Score 2
I can perform all operations with rational expressions as long as the problems are not
complex. I can graph rational basic equations, but struggle with more complex ones and
finding the asymptotes.
Score 1
With help, I can solve these problems as long as they are not too complex.
Score 0
Even if I have help, I cannot do these problems….YET!
2
6.1 Multiply and Divide Rational Expressions
Vocabulary:

Simplified form of a rational expression: A rational expression in which its
numerator and denominator have no common _______________________.

Excluded Values:_____________________________________________
Simplifying Rational Expressions and State the Excluded Values:
1.
20 x 4
15 x 3
4.
x2  7x
3x 2
2.
x 2  2 x  15
x2  9
3.
5.
3
2 x 2  15 x  18
x 2  2 x  24
4 x 2  28 x  40
x2  4
Multiplying Rational Expressions:
6.
48 x 5 y 3 x 2 y
 3 2
y4
6x y
8.
7.
4( x  5) x( x  1)

2( x  5)
x2
3x  3x 2 x 2  x  20

3x
x 2  4x  5
x 2  3x  10
 ( x 2  10 x  21)
9. 2
x  2 x  15
Dividing Expressions:
4
Dividing Rational Expressions:
10.
8 x 2 y 2 z 10 xy
 4
xz3
x z
11.
Summary
5
2x 2  7x  4 x 2  4x
 2
6 x  42
x  49
6.2 Add and Subtract Rational Expressions
Add Or Subtract:
1.
7
3

4x 4x
2.
2x
5

x6 x6
To add or subtract two rational expressions with unlike denominators, find a
_____________________ __________________________.
Find the LCD:
3.
7
2x
,
x2 x5
4.
7
2x
,
3 x 3( x  2)
5.
7
2x
, 2
(hint: factor denominators)
2
9 x 3x  3x
Add Or Subtract:
6.
8
5

2
4x
3x
7.
6
9
2x

x  3 x 1
3 x2
 2
x
8. x
10.
9.
x
x 1
 2
x  9 x  6x  9
2
3x
 2
x2 x 4
11.
2
Summary
7
x  2 2 5x
 
x  4 x 3x
6.3 Solve Rational Equations
Vocabulary:


To solve a rational equation when each side is a single rational expression (a
proportion), you can use ___________________ __________________________.
Extraneous solution: An apparent solution that must be rejected because it does
not satisfy the original equation.
Solve and check answer(s):
1.
x
9

4 4x
2.
x3
x

x5 x2
Solve rational equations that are not proportions, and check answer(s):
You can solve these easily by multiplying each side of the equation by the _______________ of
each rational expression.
9 1 4
 
3.
3x 6 3x
4.
5.
4
2x  2
1 
x
x
6
8x 2
4x
 2

x3 x 9 x3
8
6.
18
6
5


x  3x x  3 x
2
Summary
6.4 Graph Simple Rational Functions
Vocabulary:

A Rational Function is a function of the form _________________ where
p(x) and q(x) are ___________________and q(x) ≠___.
9
Graph Rational Functions:
1. y 
4
x
Steps to Graph Rational Functions
1st: Draw the asymptotes x = _______ and
y =_______.
2nd: Plot points to L and R of vertical asymptote
(at least two on each side)
3rd: Draw the branches
4th: State the domain and range.
Domain ________________________
Range _________________________
1. y 
3
x
Steps to Graph Rational Functions
1st: Draw the asymptotes x = _______ and
y =_______.
2nd: Plot points to L and R of vertical asymptote
(at least two on each side)
3rd: Draw the branches
4th: State the domain and range.
Domain ________________________
Range _________________________
10
3. y 
6
2
x3
Steps to Graph Rational Functions
1st: Draw the asymptotes x = _______ and
y =_______.
2nd: Plot points to L and R of vertical asymptote
3rd: Draw the branches
4th: State the domain and range.
Domain ________________________
Range _________________________
11
4. y 
3
1
x2
1st: Draw the asymptotes x = _______ and
y =_______.
2nd: Plot points to L and R of vertical asymptote
3rd: Draw the branches
4th: State the domain and range.
Domain ________________________
Range _________________________
5. y 
2
3
x 1
1st: Draw the asymptotes x = _______ and
y =_______.
2nd: Plot points to L and R of vertical asymptote
3rd: Draw the branches
4th: State the domain and range.
Domain ________________________
Range _______________________
12
Summary:
8. If the rational function is in the form y 
a
 k , where is the horizontal asymptote? Vertical
xh
asymptote?
6.5 Graph General Rational Functions
Analyzing Graphs:
Identify the x-intercept(s), vertical asymptote(s), and horizontal asymptote(s) of the graph of the
function.
5
x2  9
1. y  2
2. y  2
x 1
x  2 x  15
13
Graphing Functions:
3. y 
8
x  x6
Steps to Graph Rational Functions
2
1st: Draw the asymptotes x = _______ and
y =_______.
2nd: Plot x-intercepts
3rd: Plot points to L and R of vertical asymptote
4th: Draw the branches
5th: State the domain and range.
Domain ________________________
Range _________________________
4. f ( x) 
x 2  2x  3
x2
1st: Draw the asymptotes x = _______ and
y =_______.
2nd: Plot x-intercepts
3rd: Plot points to L and R of vertical asymptote
4th: Draw the branches
5th: State the domain and range.
Domain ________________________
Range _________________________
14
5. y 
x 2  16
2x 2  1
1st: Draw the asymptotes x = _______ and
y =_______.
2nd: Plot x-intercepts
3rd: Plot points to L and R of vertical asymptote
4th: Draw the branches
5th: State the domain and range.
Domain ________________________
Range _________________________
6. f ( x) 
x2  3
2x 3
1st: Draw the asymptotes x = _______ and
y =_______.
2nd: Plot x-intercepts
3rd: Plot points to L and R of vertical asymptote
4th: Draw the branches
5th: State the domain and range.
Domain ________________________
Range _________________________
15
7. y 
4x  2
x 1
1st: Draw the asymptotes x = _______ and
y =_______.
2nd: Plot x-intercepts
3rd: Plot points to L and R of vertical asymptote
4th: Draw the branches
5th: State the domain and range.
Domain ________________________
Range _________________________
Summary
8. Explain how to find the x-intercepts of a rational function of the form f ( x) 
p ( x)
.
q ( x)
p ( x)
has any horizontal
q ( x)
asymptotes. Explain when a function has one versus does not have one.
9. Explain how you would determine if a function of the form f ( x) 
10. Explain how you would determine if a function of the form f ( x) 
asymptotes.
16
p ( x)
has any vertical
q ( x)