ALGEBRA II
CHAPTER 9 NOTES
RATIONAL EQUATIONS AND FUNCTIONS
Name __________________________________________________________
Algebra II – Chapter 9 Notes
Algebra II – Chapter 9 Notes
Algebra II
9.2 Graphing Simple Rational Functions Day One
Today I am…graphing simple rational functions.
I am successful today when I can…graph simple rational functions.
It is important for me to know/do this because…graphs of simple rational functions can help solve real-life problems.
1. Graph the function
y
1
x
x-intercept: __________
y-intercept: __________
vertical asymptote: ___________
horizontal asymptote: __________
domain: __________
range: __________
Asymptote: a line that is approached by the graph.
The distance between the graph and the line approaches zero.
1
a
k.
is called a HYPERBOLA. It is a rational function in the form of y 
x
xh
● Hyperbolas have asymptotes of x  h and y  k .
● To find the y-intercept, plug 0 in for x and solve for y.
● To find the x-intercept, plug 0 in for y and solve for x.
The graph of
y
● Plot a few more points on each side of the vertical asymptote to get a graph. The graph will have two branches.
2. Graph the function
y
3
x2
x-intercept: __________
y-intercept: __________
vertical asymptote: ___________
horizontal asymptote: __________
domain: __________
range: __________
Algebra II – Chapter 9 Notes
3. Graph the function
y
1
3
x 1
x-intercept: __________
y-intercept: __________
vertical asymptote: ___________
horizontal asymptote: __________
domain: __________
range: __________
4. Graph the function
y
1
3
x4
x-intercept: __________
y-intercept: __________
vertical asymptote: ___________
horizontal asymptote: __________
domain: __________
range: __________
Homework: 9.2 Day One Worksheet
Algebra II – Chapter 9 Notes
Algebra II
9.2 Graphing Simple Rational Functions Day Two
Today I am…graphing simple rational functions.
I am successful today when I can…graph simple rational functions.
It is important for me to know/do this because…graphs of simple rational functions can help solve real-life problems.
ax  b
. These graphs are also called hyperbolas.
cx  d
● The vertical asymptote occurs at the x-value that makes the denominator zero. So, solve the denominator for zero.
a
● The horizontal asymptote is the line y  .
c
● To find the y-intercept, plug 0 in for x and solve for y.
● To find the x-intercept, solve for the numerator for zero.
Rational functions can also take the form
y
● Plot a few more points on each side of the vertical asymptote to get a graph. The graph will have two branches.
1. Graph the function
y
x2
3x  3
x-intercept: __________
y-intercept: __________
vertical asymptote: ___________
horizontal asymptote: __________
domain: __________
range: __________
2. Graph the function
y
2x
x4
x-intercept: __________
y-intercept: __________
vertical asymptote: ___________
horizontal asymptote: __________
domain: __________
range: __________
Algebra II – Chapter 9 Notes
3. Graph the function
y
x  2
x 5
x-intercept: __________
y-intercept: __________
vertical asymptote: ___________
horizontal asymptote: __________
domain: __________
range: __________
Homework: 9.2 Day Two Worksheet
Algebra II – Chapter 9 Notes
Algebra II
9.3 Graphing General Rational Functions
Today I am…graphing general rational functions.
I am successful today when I can…graph general rational functions.
It is important for me to know/do this because…graphs of general rational functions can help solve real-life problems.
Graphs of Rational Functions
p  x
am x m  ...
f  x 

q  x  bn x n  ...
• The x-intercepts (where graph crosses x-axis) are the real zeros of p(x)…
.
• The graph of f has a vertical asymptote at each real zero of q(x)…
.
• The graph of f has at most one horizontal asymptote
…if m < n (
), the line y  0 (x-axis) is a horizontal asymptote.
y
am
is a horizontal asymptote.
bn
=n(
…if m
(Reduce the coefficients in front of the 2 variables)
>n(
), the graph has no horizontal asymptote.
), the line
The graph’s end behavior is the same as the graph of
1.
y
…if m
x
2
x 4
2.
y
y
am m n
x .
bn
x2  4
x2  1
x-intercept: __________
x-intercept: __________
y-intercept: __________
y-intercept: __________
vertical asymptote: ___________
vertical asymptote: ___________
horizontal asymptote: __________
horizontal asymptote: __________
Algebra II – Chapter 9 Notes
x2  x  6
3. y 
x2
x-intercept: __________
8x2
4. y 
4 x2  9
x-intercept: __________
y-intercept: __________
y-intercept: __________
vertical asymptote: ___________
vertical asymptote: ___________
horizontal asymptote: __________
horizontal asymptote: __________
Homework: 9.3 Worksheet
Algebra II – Chapter 9 Notes
Algebra II
Factoring Review for 9.4 Notes
1. x  6 x  8 ____________________
2. x  9 x  8 ____________________
3. x  2 x  8 ____________________
4. x  7 x  8 ____________________
5. x  9 ____________________
6. 25 x  16 ____________________
2
2
2
2
2
2
7. x  8 ________________________________________
3
8. 27 x  64 ________________________________________
3
9. 6 x  11x  4 ____________________
2
Homework: Factoring Review worksheet
Algebra II – Chapter 9 Notes
Algebra II – Chapter 9 Notes
Algebra II
9.4 Multiplying and Dividing Rational Expressions
Today I am…multiplying and dividing rational expressions.
I am successful today when I can…multiply and divide rational expressions.
It is important for me to know/do this because…rational expressions can model real-life quantities.
Simplifying rational expressions requires two steps:
• Factor the numerator and denominator (if possible)
• Divide/cancel out any factors they have in common
1.
9 x5 6 5 x
 
5 x 7 18
2.
6 x 2 y 3 10 x3 y 4

2 x 2 y 2 18 y 2
3.
9 x3 y 12 x 4 y 5

3x 2 y 3 27 y 2
4.
x2  5x  6
x2  1
5.
2 x3  14 x 2  36 x
2 x 2  16 x  18
6.
27 x3  8
3x 2  13x  10
7.
x2  9
x2  4

x2  4 x  4 x2  x  6
8.
x 3
 16 x 2  4 x  1
64 x3  1
Algebra II – Chapter 9 Notes
3
x 2  3x
9.

4 x  8 x2  x  6
x5  4 x3 x5  x 4  2 x3
10. 2

x x2
x2  1
x
16 x 2  1
11.
  4 x  1 
x3
x3
Homework: 9.4 Day One Worksheet
Algebra II – Chapter 9 Notes
Algebra II
9.5 Day 1 Addition, Subtraction, and Complex Numbers
Today I am…adding and subtracting rational expressions and simplifying complex fractions.
I am successful today when I can…add and subtract rational expressions.
It is important for me to know/do this because…rational expressions can model real-life quantities.
***IN ORDER TO ADD/SUBTRACT FRACTIONS, YOU MUST HAVE A COMMON DENOMINATOR***
1.
1 5

18 18
5.
3x  1
6

x4 x4
8.
6x
4x

3x  1 2 x  5
2.
4
5

3x 3x
6.
3.
5 1

12 8
2x 1
x

x 1 x 1
9.
4.
7.
4
x
 3
3
3x 6 x  3x 2
Algebra II – Chapter 9 Notes
x
8
 2
x2 x 4
2x
4

x3 x3
10.
x 1
2
 2
x  4x  4 x  4
2
11.
x 1
1
 2
x  6x  9 x  9
2
Homework: 9.5 Day One Worksheet
Algebra II – Chapter 9 Notes
Algebra II
9.5 Day 2 Addition, Subtraction, and Complex Numbers
Today I am…adding and subtracting rational expressions and simplifying complex fractions.
I am successful today when I can…add and subtract rational expressions.
It is important for me to know/do this because…rational expressions can model real-life quantities.
***IN ORDER TO ADD/SUBTRACT FRACTIONS, YOU MUST HAVE A COMMON DENOMINATOR***
1
1. 2
3
4
2
x 1
4.
4
1

x 1 x
2.
x
3
3. 2
1
1
x
1
1 1

2 3
5.
3
x4
1
3

x  4 x 1
Homework: 9.5 Day Two Worksheet
Algebra II – Chapter 9 Notes
Algebra II – Chapter 9 Notes
Algebra II
9.6 Solving Rational Equations
Today I am…solving rational equations.
I am successful today when I can…solve rational equations.
It is important for me to know/do this because…rational equations can help solve real-life problems.
***Be careful when solving…you may get extraneous solutions. Plug your answer back in to the original
equation to make sure it works and WATCH THAT YOU DO NOT GET DIVISION BY ZERO.***
1.
x
1

x6 x4
3.
5x
10
7
x2
x2
________________
__________
2.
4.
3 1 12
 
x 2 x
2
1

x  4x x  4
_________________
2
__________
Algebra II – Chapter 9 Notes
5.
5
5
__________
 4
x 1
x 1
x
2x
18
6.

 2
x 3 x 3 x 9
___________
3x
2x
5 x 2  15 x  20
7.
____________


x 1 x  6
x2  7 x  6
Homework: 9.6 Day One Worksheet
Algebra II – Chapter 9 Notes