Name: _______________________
Period: ______________________
Section 9.1 Variation
Objective(s): Solve a problem by applying inverse and joint variation.
Essential Question: Suppose x varies inversely with y and y varies inversely with z. How does x vary with
z? Justify your answer algebraically.
Homework: Assignment 9.1 #1 – 28 in the homework packet.
Notes:
Direct Variation
Two variables x and y show direct variation if y = kx. k is called the constant of variation. We say y varies
directly as x.
What this means in English… as one quantity goes up, so does the other.
For example, the number of calories a person burns while exercising varies directly with the length of
time they exercise. So, if you increase your exercise time from 30 minutes to 45 minutes, you will burn
more calories.
Inverse Variation
Two variable x and y show inverse variation if y = k/x. We say y varies inversely as x.
As one quantity goes UP, the other goes DOWN.
For example, the number of hours it takes for a block of ice to melt varies inversely with room
temperature. As room temperature increases, it takes less time for the ice to melt.
Joint Variation
Joint variation occurs when a quantity varies directly with two or more other quantities. We say z varies
jointly as x and y and the equation would be z = kyx.
For example, work done when lifting an object varies jointly with the mass of the object and the height
that the object is lifted.
Reflection:
1
Translating Problems into an Equation: 3 types
varies directly as

= k
varies jointly as

= k
and

varies inversely as
=
k
Write an equation to describe the variation.
Example 1:
w varies directly as v
Example 2:
r varies directly as t
Example 3:
m varies inversely as n
Example 4:
t varies inversely as s
Example 5:
x varies jointly as y and z
Example 6:
w varies jointly as p and q
Example 7:
J varies directly as M and inversely as N
Example 8:
i varies inversely as the square of d
Example 9:
D varies directly with the square root of L and inversely with the square of T
Reflection:
2
Example 10:
the handle.
The force F needed to loosen a bolt with a wrench varies inversely with the length l of
Example 11:
The gravitational force F between two objects varies jointly with their masses m1 and m2
and inversely with the square of the distance d between the two objects.
Steps for Variation Problems
1.
2.
3.
4.
5.
6.
Write an equation to describe the variation.
Substitute the first set of numbers into the equation.
Solve for k.
Replace k in your equation with the number you just found.
Substitute the second set of numbers into the equation (the one where k is a number).
Solve for whatever variable is asked for.
Solve. Find k and then the answer.
Example 12:
If r varies directly with q, and r = 56 when q = 7, find r when q is 6.
Variation equation: ___________________
r = ________
q = ________ (the FIRST q)
Substitute r and q into variation equation: ___________________ and solve for k.
k = ________
Use the variation equation and substitute with k and the new variables: ____________________
q = ________ (the SECOND q)
Solve for r.
r = ________
Example 13:
If n varies directly with m, and n = 72 when m = 8, find n when m is 5.
Variation equation: ______________________
n = ________
Reflection:
3
Example 14:
If y varies directly as x and inversely as z, and y = 1 when x = 9 and z = 18, find y when x
is 2 and z is 3.
Variation equation: ______________________
y = ________
Example 15:
3.
If w varies jointly as p2 and r, and w = 81 when p = 3 and r = 3, find w when p is 2 and r is
Variation equation: ______________________
w = ________
Example 16:
The intensity I (in watts per square meter) of a sound varies inversely with the square of
the distance d (in meters) from the sounds source. At a distance of 1 meter from the stage, the intensity
of the sound at a rock concert is about 10 watts per square meter. If you are sitting 15 meters back from
the stage, what is the intensity of the sound you hear?
Reflection:
4
Sample CCSD Common Exam Practice Question(s):
1. The value of s varies jointly with m and p. If s = 10 when m = 3 and p = 4, what is the value of s
when m = 7 and p = 2?
A.
47
6
B.
81
5
C.
49
5
D.
35
3
Reflection:
5
Section 9.2 Simplifying Rational Functions
Objective(s): Simplify rational expressions.
Essential Question: What is the formula for factoring sum/difference of two cubes?
Homework: Assignment 9.2 #29 – 41 in the homework packet.
Notes:
Simplifying Rational Expressions
1. Factor numerator and denominator
2. Cancel common FACTORS (not terms – leave anything INSIDE the parentheses ALONE)
Simplify the rational expression.
Example 1:
( x  1)( x  8)
( x  2)( x  1)
Example 3:
20 x 2  12 x
20 x
Example 2:
( x  3)( x  5)
( x  5)( x  3)
Hint: What is the GCF?
Do NOT cancel 20x on top with the 20x on the bottom!!!!
Example 4:
Reflection:
5 x  30
4 x  24
Example 5:
12 x  12 y
x y
6
Example 6:
9 x 2  27 x3
5 x  15 x 2
Example 7:
2x  3
10 x  19 x  6
Example 8:
x2  6 x  8
x 2  10 x  16
Example 9:
3x3  27 x
x3  2 x 2  15 x
Example 10:
x3  2 x 2  8 x
2 x3  32 x
Reflection:
2
7
Section 9.3 Multiplying and Dividing Rational Functions
Objective(s): Multiply and divide rational expressions.
Essential Question: What is the formula for factoring sum/difference of two cubes?
Homework: Assignment 9.3 #42 – 54 in the homework packet.
Notes:
Multiplying Fractions
1.
2.
3.
4.
5.
Factor numerator and denominator
Cancel common FACTORS
Multiply numerators
Multiply denominators
Leave in FACTORED form (do not distribute/FOIL)
Example 11:
2 5
3 2
Example 12:
2 x 2 15
5 x3
Example 14:
40 xy 2 8x  80
x 2  100 5x 2 y 2
Reflection:
Example 13:
8x  8 7 x2
x
9x  9
8
Example 15:
x2  5x  6
x2  5x
x 2  7 x  10 x 2  7 x  12
Dividing Fractions
1.
2.
3.
4.
5.
6.
Flip the SECOND fraction and change the sign to multiply
Factor numerator and denominator
Cancel common FACTORS
Multiply numerators
Multiply denominators
Leave in FACTORED form (do not distribute/FOIL)
Example 16:
2 8

9 3
Example 18:
x 2  14 x  49 x 2  49

x7
7 x  49
Reflection:
Example 17:
2 x 2 x3

3
12
9
x 2  12 x  35
x2  7 x

x 2  14 x  45 x 2  18 x  81
Example 19:
Sample CCSD Common Exam Practice Question(s):
1. Simplify the expression:
x2  3 x  4 x2  4 x
 2
x 2  16
x  x2
A. 1
B.
1
x4
C.
x2
x4
D.
x
x2
Reflection:
10
Section 9.4 Solving Rational Equations
Objective(s): Solve equations involving rational expressions.
Essential Question: When is cross-multiplying an appropriate method for solving a rational equation?
Homework: Assignment 9.4 #55 – 67 in the homework packet.
Notes:
Finding the Least Common Multiple
1. Factor the expression (if it is a binomial or trinomial)
2. List each factor to the HIGHEST exponent
Find the least common multiple.
Example 1:
2
5
Example 2:
4
6
LCM = _________________
LCM = _________________
Example 3:
Example 4:
x
4x
LCM = _________________
Example 5:
x2 – 4
LCM = _________________
Reflection:
6
30
5x
LCM = _________________
x+2
Example 6:
x2 + 5x
x2 – 25
LCM = _________________
11
Solving Rational Expressions
1. Find the LCM using all of the denominators.
2. Multiply every NUMERATOR by the LCM.
3. Cancel each denominator with your LCM. (If you still have something in the denominator, then
you did something wrong.)
4. Solve for x.
Solve the equation.
Example 7:
x x
 2
3 7
LCM = _________________
Example 8:
3 x 1 8


2x
x
x
LCM = _________________
Example 9:
1 7 3
 
4x 8 x
LCM = _________________
Reflection:
12
Example 10:
1
1
3


3x 5 x
15
LCM = _________________
Example 11:
3x  1 3

x
4
LCM = _________________
Example 12:
5
7

x  36 x  6
LCM = _________________
Example 13:
2
6

x  9x x  9
LCM = _________________
Reflection:
2
2
13
John can paint a birdhouse in 2 hours and Kathy can complete the same job in 3 hours. How long would
it take for them to complete the job if they were working together?
Why is the answer NOT 5 hours?????
“Work" problems involve situations such as two people working together to paint a house. You are
usually told how long each person takes to paint a similarly-sized house, and you are asked how long it
will take the two of them to paint the house when they work together.
x
x

 1 job
time for person 1 time for person 2
Example 14:
John can paint a birdhouse in 2 hours and Kathy can complete the same job in 3 hours.
How long would it take for them to complete the job if they were working together?
Example 15:
Maria can sew a quilt in 3 hours and Sam can sew the same quilt in 6 hours. If Maria and
Sam work together on the job and the cost of labor is $30 per hour, what should the labor estimate be?
Reflection:
14
Sample CCSD Common Exam Practice Question(s):
1.
2.
Solve
7
1
1
.

3n 4n
A.
n
1
12
B.
n
2
49
C.
n
1
42
D.
n
1
84
Solve the rational equation for y:
5
5 y  10
5
 2

y  5 y  8 y  15 y  3
A. y = –2
B.
y=0
C.
y=2
D. y = 4
3.
The equivalent resistance R of two resistors in parallel, with resistances
r1 and r2 , is given by the
formula:
1 1 1
 
R r1 r2
If
r1 has twice the resistance of r2 , what is the value of r2 if the equivalent resistance is 30 ohms?
A. 15 ohms
B.
30 ohms
C.
45 ohms
D. 90 ohms
Reflection:
15