Download Inverse and Joint Variation

9.1
Inverse and Joint Variation
Goals p Write and use inverse variation models.
p Write and use joint variation models.
Your Notes
VOCABULARY
Inverse variation Two variables x and y show inverse
k
variation provided y where k is a nonzero
x
constant.
Constant of variation The nonzero constant in a direct
variation equation, inverse variation equation, or joint
variation equation
Joint variation A relationship that occurs when a
quantity varies directly as the product of two or
more other quantities
Example 1
Classifying Direct and Inverse Variation
Tell whether x and y show direct variation, inverse variation,
or neither.
Given Equation
Rewritten Equation
a. y x 2
Type of Variation
Neither
b. 2 yx
2
y x
Inverse
y
c. x 7
y 7x
Direct
Checkpoint Tell whether x and y show direct variation,
inverse variation, or neither.
1. yx 1
inverse
variation
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Algebra 2 Notetaking Guide • Chapter 9
y
2. x
1.3
direct
variation
3. y x 1
neither
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Your Notes
Writing an Inverse Variation Equation
Example 2
The variables x and y vary inversely, and y 2 when x 3.
a. Write an equation that relates x and y.
b. Find y when x 2.
Solution
a.
k
y x
k
2 3
6 k
Write general equation for inverse variation.
Substitute for y and for x.
Solve for k.
6
The inverse variation equation is y .
x
b. When x 2, the value of y is:
6
y 3
2
Checking Data for Inverse Variation
Example 3
Tell whether the following data show inverse variation. If they
do, find a model for the relationship between a and b.
a
5
10
15
20
25
b
21
10.5
7
5.25
4.2
Solution
Each product ab is equal to 105 . For instance,
(5)(21) 105 and (20)(5.25) 105 . So, the
data do show inverse variation. A model for the
105
relationship is b .
a
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Your Notes
Comparing Different Types of Variation
Example 4
Write an equation for the given relationship.
Relationship
Equation
a. v varies directly with u.
v ku
b. v varies inversely with u.
k
v u
c. w varies jointly with t, u, and v.
w ktuv
d. v varies inversely with the cube of u.
k
v u3
e. w varies directly with u and inversely with v.
ku
w v
Checkpoint Complete the following exercises.
4. The variables x and y vary inversely, and y 3.5 when
x 4. Find an equation that relates x and y. Find y when
x 6.
14 7
y ; x 3
5. Do the data below show inverse variation? If so, find a
model for the relationship between x and y.
x
2
4
6
8
10
y
12
6
4
3
2.4
24
yes; y x
Homework
6. Write an equation for the given relationship: y varies
jointly with x and z and varies inversely with the square
of v.
kxz
y v2
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Algebra 2 Notetaking Guide • Chapter 9
9.2
Graphing Simple
Rational Functions
Goals p Graph simple rational functions.
p Use the graph of a rational function to solve real-life
problems.
Your Notes
VOCABULARY
p(x)
Rational function A function of the form f(x) q(x)
where p(x) and q(x) are polynomials and q(x) 0
Hyperbola The set of all points P such that the
difference of the distances from P to two fixed points,
called the foci, is constant
Branches of a hyperbola Two symmetrical parts of a
hyperbola
Example 1
Graphing a Rational Function
3
Graph y 1. State the domain and range.
x2
Solution
Draw the asymptotes x 2
and y 1 .
y
5
(3, 4)
Plot two points to the left of
the vertical asymptote, such
as (1, 0 ) and (1, 2 ), and
two points to the right, such as
(3, 4 ) and (5, 2 ).
Use the asymptotes and plotted
3
(5, 2)
(1, 0)
1
1
1
1
3
5
x
(1, 2)
points to draw the branches of
the hyperbola.
The domain is all real numbers except 2 , and the range
is all real numbers except 1 .
Lesson 9.2 • Algebra 2 Notetaking Guide
207
Your Notes
Example 2
Graphing a Rational Function
4x 2
Graph y . State the domain and range.
2x 2
Solution
Draw the asymptotes. Solve 2x 2 0 for x to find the
vertical asymptote x 1 . The horizontal asymptote is
a
4
y 2 .
c
2
y
Plot two points to the left of
the vertical asymptote, such
as (4, 3 ) and (2, 5 ),
and two points to the right,
such as (0, 1 ) and (2, 1 ).
Use the asymptotes and plotted
(2, 5)
5
(4, 3)
3
1
5
3
1
1
points to draw the branches of
the hyperbola.
(2, 1)
1
3 x
(0, 1)
The domain is all real numbers except 1 , and the range
is all real numbers except 2 .
Checkpoint Complete the following exercise.
2
1. Graph y 1.
x2
State the domain and range.
y
3
(3, 1)
(4, 0)
5
3
1
1
1
3
(1, 3)
domain: all real numbers except 2;
range: all real numbers except 1
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Algebra 2 Notetaking Guide • Chapter 9
5
1
(0, 2)
x
Your Notes
Example 3
Writing a Rational Model
You are arranging a dinner at a local restaurant. The cost to
rent a dining room is $300. In addition to this one-time
charge, the unit cost of each plate is $40.
a. Write a model that gives the average cost per person as a
function of the number of people attending.
b. Graph the model and use it to estimate how many people
must attend to drop the average cost to $45 per person.
c. Describe what happens to the average cost as the number
of people attending increases.
Solution
a. The average cost is the total cost of the dinner divided by
the number of people attending.
One-time Unit p Number
charges
cost
attending
Verbal
Model
Average cost
Labels
Average cost A
(dollars)
One-time charges 300
(dollars)
Unit cost 40
(dollars)
Number attending x
(people)
Number attending
Homework
b. The A-axis is the vertical
asymptote and the line
A 40 is the horizontal
asymptote. The domain is
x > 0 and the range is
A > 40 . When A 45, the
value of x is 60 . So, you
need at least 60 people to
attend for the average cost to
drop to $45 per person.
Average cost (dollars)
300 40x
Algebraic A x
Model
A
70
60
50
A
300 40x
x
(60, 45)
40
30
20
10
0
0 10 20 30 40 50 60 70 80 x
Number attending
c. As the number of people attending increases, the average
cost per person gets closer and closer to $40 .
Lesson 9.2 • Algebra 2 Notetaking Guide
209
9.3
Graphing General
Rational Functions
Goals p Graph general rational functions.
p Use the graph of a rational function to solve real-life
problems
Your Notes
GRAPHS OF RATIONAL FUNCTIONS
Let p(x) and q(x) be polynomials with no common factors
other than 1. The graph of the rational function
am x m ⫹ am ⫺ 1x m ⫺ 1 ⫹ . . . ⫹ a1x ⫹ a0
p(x)
f(x) ⫽ ᎏᎏ ⫽ ᎏᎏᎏᎏᎏ
bn x n ⫹ bn ⫺ 1x n ⫺ 1 ⫹ . . . ⫹ b1x ⫹ b0
q(x)
has the following characteristics.
1. The x-intercepts of the graph of f are the real zeros
of p(x) .
2. The graph of f has a vertical asymptote at each real zero
of q(x) .
3. The graph of f has at most one horizontal asymptote.
• If m < n, the line y ⫽ 0 is a horizontal asymptote.
a
• If m ⫽ n, the line y ⫽ ᎏm
ᎏ is a horizontal asymptote.
bn
• If m > n, the graph has no horizontal asymptote .
The graph’s end behavior is the same as the graph of
a m⫺n
y ⫽ ᎏm
ᎏx
.
bn
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Your Notes
Example 1
Graphing a Rational Function (m < n)
12
Graph y . State the domain and range.
x4 4
Solution
The numerator has no zeros, so there is no x-intercept .
The denominator has no real zeros, so there is no vertical
asymptote . The degree of the numerator ( 0 ) is less than
the degree of the denominator ( 4 ), so the line y 0 is a
horizontal asymptote.
y
The graph passes through the
points (2, 0.6 ), (1, 2.4 ),
(0, 3 ), (1, 2.4 ), and (2, 0.6 ).
The domain is all real numbers ,
and the range is 3 ≤ y < 0 .
3
1
3
1
1
1
3
x
3
5
Checkpoint Complete the following exercise.
10
.
1. Graph y 3
x 8
State the domain and range.
y
3
1
3
1
1
1
3 x
domain: all real numbers except 2;
range: all real numbers except 0
Lesson 9.3 • Algebra 2 Notetaking Guide
211
Your Notes
Example 2
Graphing a Rational Function (m ⴝ n)
x2 ⴚ 4
Graph y ⴝ ᎏ
ᎏ.
x2 ⴚ 1
The numerator can be factored as (x ⫹ 2)(x ⫺ 2) , so the
x-intercepts of the graph are ⫺2 and 2 . The denominator
can be factored as (x ⫹ 1)(x ⫺ 1) , so the denominator has
zeros ⫺1 and 1 . This implies that the lines x ⫽ ⫺1 and
x ⫽ 1 are vertical asymptotes of the graph. The degree of
the numerator ( 2 ) is equal to the degree of the denominator
a
( 2 ), so the horizontal asymptote is y ⫽ ᎏm
ᎏ ⫽ 1 . To draw
bn
the graph, plot points between and beyond the vertical
asymptotes.
To the left of x ⴝ ⴚ1
Between x ⴝ ⴚ1
and x ⴝ 1
To the right of x ⴝ 1
y
x
y
⫺4
0.8
5
⫺2
0
3
⫺0.5
5
1
0
4
0.5
5
2
0
4
0.8
⫺3
⫺1
⫺1
1
3
Checkpoint Complete the following exercise.
⫺x 3
ᎏ.
2. Graph y ⫽ ᎏ
x3 ⫺ 1
y
3
1
⫺3
⫺1
⫺1
⫺3
⫺5
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1
3
x
x
Your Notes
Example 3
Graphing a Rational Function (m > n)
x2 ⴚ 5x ⴙ 4
Graph y ⴝ ᎏᎏ.
xⴚ5
The numerator can be factored as (x ⫺ 4)(x ⫺ 1) , so the
x-intercepts of the graph are 4 and 1 . The only zero of the
denominator is 5 , so the only vertical asymptote is x ⫽ 5 .
The degree of the numerator ( 2 ) is greater than the degree
of the denominator ( 1 ), so there is no horizontal
asymptote and the end behavior of the graph of f is the
same as the end behavior of the graph of y ⫽ x 2 ⫺ 1 ⫽ x .
To draw the graph, plot points to the left and right of the
vertical asymptote.
To the left of x ⴝ 5
To the right of x ⴝ 5
x
y
0
⫺0.8
1
0
3
1
4
0
6
10
7
9
9
10
y
14
10
6
2
⫺2
⫺2
2
6
10
14 x
Checkpoint Complete the following exercise.
x2 ⫺ x ⫺ 5
3. Graph y ⫽ ᎏᎏ.
x⫺2
y
5
3
Homework
1
⫺3
⫺1
⫺1
1
3
5 x
⫺3
Lesson 9.3 • Algebra 2 Notetaking Guide
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9.4
Multiplying and Dividing
Rational Expressions
Goals p Multiply and divide rational expressions.
p Model real-life quantities using rational expressions.
Your Notes
VOCABULARY
Simplified form of a rational expression A rational
expression in which the numerator and denominator
have no common factors (other than 1)
SIMPLIFYING RATIONAL EXPRESSIONS
Let a, b, and c be nonzero real numbers or variable
expressions. Then the following property applies:
ac
a
Divide out common factor c.
bc
b
Example 1
Simplifying a Rational Expression
x2 ⴚ 2x ⴚ 3
Simplify: x2 ⴚ 9
x2 2x 3
(x 3)(x 1)
x2 9
(x 3)(x 3)
x1
x3
Factor numerator
and denominator.
Divide out common factor
x 3 and write
simplified form.
Checkpoint Complete the following exercise.
x2 2x 15
1. Simplify: x2 6x 5
x3
x1
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Algebra 2 Notetaking Guide • Chapter 9
Your Notes
Example 2
Multiplying Rational Expressions
3x2 6x
x4
Multiply: p
2
2
x 4x
x 9x 14
Solution
3x2 6x
x4
p
2
2
x 4x
x 9x 14
3x(x 2)
x4
p x(x 4)
(x 2)(x 7)
Factor numerators
and denominators.
3x(x 2)(x 4)
x(x 4)(x 2)(x 7)
Multiply numerators
and denominators.
3
x7
Example 3
Divide out common
factors and write
simplified form.
Multiplying by a Polynomial
2x
Multiply: p (x2 x 1)
3
x 1
Solution
2x
p (x2 x 1)
3
x 1
2x
x2 x 1
p
x3 1
1
Write polynomial as
rational expression.
2x(x2 x 1)
(x 1)(x2 x 1)
Factor and multiply.
2x
x1
Divide out common factor and
write simplified form.
Lesson 9.4 • Algebra 2 Notetaking Guide
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Your Notes
Checkpoint Multiply the rational expressions.
x3
4x2y3
p
2. xy2
x2 x 12
4xy
x4
Example 4
x
3. p (x2 2x 4)
3
x 8
x
x2
Dividing Rational Expressions
x2 4x 3
x1
Divide: 2
4x 12x
3
Solution
x2 4x 3
x1
2
4x 12x
3
x2 4x 3
3
p 2
4x 12x
x1
Multiply by reciprocal.
(x 3)(x 1)
3
p 4x(x 3)
x1
Factor.
3
4x
Divide out common factors
and write simplified form.
Checkpoint Divide the rational expressions.
20x2 x 1
4x 1
4. 3
7x x
x
x3 4x
5. (x4 16)
2
Homework
5x 1
7x2 1
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x
2
2(x 4)
9.5
Addition, Subtraction, and
Complex Fractions
Goals p Add and subtract rational expressions.
p Simplify complex fractions.
Your Notes
VOCABULARY
Complex fraction A fraction that contains a fraction in
its numerator or denominator
Example 1
Subtracting with Like Denominators
7
11 4
11
4
1
7x
7x
x
7x
7x
Example 2
Subtract
numerators
and simplify.
Adding with Unlike Denominators
xⴙ2
x
Add: ⴙ 2
4x
3x ⴙ 9x
First find the least common denominator.
Factor the denominators: 4x 4 p x
3x2 9x 3 p x p (x 3)
The LCD is 12x(x 3) . Use this to rewrite each expression.
x
x2
x
x2
2
4x
4x
3x 9x
3x(x 3)
(x 2) [3(x 3)]
4x [3(x 3)]
x (4)
3x(x 3)(4)
3x 2 15x 18
4x
12x(x 3)
12x(x 3)
3x 2 19x 18
12x(x 3)
Lesson 9.5 • Algebra 2 Notetaking Guide
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Your Notes
Example 3
Subtracting with Unlike Denominators
2x
x2
Subtract: 2
2
x 1
x 2x 1
Solution
2x
x2
2
2
x 1
x 2x 1
2x
(x 1)(x 1)
x2
(x 1)2
(x 2)(x 1)
2x(x 1)
2
(x 1)(x 1)(x 1)
(x 1) (x 1)
2x2 2x (x2 x 2)
x2 x 2
2
2
(x 1) (x 1)
(x 1) (x 1)
Checkpoint Perform the indicated operation.
2
4
1. 3x
3x
2
x
3
x
3. x2 16
x2 8x 16
x2 x 12
2
(x 4) (x 4)
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1
4
2. x5
x5
3
x5
1
5x
4. 4x2
2x3 8x2
9x 4
4x2(x 4)
Your Notes
Example 4
Simplifying a Complex Fraction
3
x1
Simplify: 1
3
x1
x
Solution
3
3
x1
x1
1
3
2x 1
x
x1
x(x 1)
Add fractions in
denominator.
x (x 1)
3
p (2x 1)
x1
Multiply by reciprocal.
3x
2x 1
Divide out common
factor and write in
simplified form.
Checkpoint Complete the following exercise.
x2
3x 1
5. Simplify: 1
2
x
3x 1
Homework
x(x 2)
x1
Lesson 9.5 • Algebra 2 Notetaking Guide
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9.6
Solving Rational Equations
Goals p Solve rational equations.
p Use rational equations to solve real-life problems.
Your Notes
VOCABULARY
Cross multiplying A method of solving a simple rational
equation for which each side of the equation is a
single rational expression. Equal products are formed
by multiplying the numerator of each expression by
the denominator of the other.
Example 1
An Equation with One Solution
4
7
1
Solve: ⴚ ⴝ 3
x
6
4
7
1
3
x
6
冢
冣
Write original equation.
冢 冣
4
7
1
6x 6x 3
x
6
8x 42 x
Multiply each side by LCD ⴝ 6x .
Simplify.
8x x 42
Add 42 to each side.
7x 42
Subtract x from each side.
x6
Divide each side by 7 .
The solution is 6 . Check this in the original equation.
Checkpoint Complete the following exercise.
7
5
1
1. Solve: 2x
3
12
2
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Algebra 2 Notetaking Guide • Chapter 9
Your Notes
Example 2
An Equation with an Extraneous Solution
xⴙ6
5x ⴙ 12
Solve: ⴝ 2 ⴚ xⴙ3
xⴙ3
The least common denominator is x 3 .
x6
5x 12
2 x3
x3
x6
5x 12
(x 3) p (x 3) p 2 (x 3) p x3
x3
x 6 2(x 3) (5x 12)
x 6 2x 6 5x 12
4x 12
x 3
The solution appears to be 3 . After checking it in the
original equation, however, you can conclude that 3 is an
extraneous solution because it leads to division by zero .
So, the original equation has no solution .
Example 3
An Equation with Two Solutions
3x ⴚ 1
12
Solve: ⴝ 2 ⴙ xⴚ2
(x ⴚ 2)(x ⴚ 1)
The LCD is (x 2)(x 1) .
3x 1
12
2 x2
(x 2)(x 1)
(x 1)(3x 1) 2(x 2)(x 1) 12
3x2 4x 1 2x2 6x 16
x2 2x 15 0
(x 5)(x 3) 0
x5 0
x 5
or
x3 0
or
x 3
The solutions are 5 and 3 .
Lesson 9.6 • Algebra 2 Notetaking Guide
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Your Notes
Checkpoint Solve the equation.
3x 2
x6
2. 3 x4
x4
2(x 3)
5
3. 1 2
x2
x 4
3, 1
no solution
Example 4
Solving an Equation by Cross Multiplying
3x 2
4
Solve: 2
x 3x
x4
Solution
3x 2
4
2
x 3x
x4
(3x 2) (x 4) 4 (x2 3x)
3x2 14x 8 4x2 12x
Write original equation.
Cross multiply.
Simplify.
0 x2 2x 8
Write in standard form.
0 (x 4)(x 2)
Factor.
x 4 or x 2
Zero product property
The solutions are 4 and 2 . Check these in the original
equation.
Checkpoint Complete the following exercise.
Homework
2x 8
x4
4. Solve: x2
x1
4
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