Transcript A.8 Direct and Inverse Variation

Write
The amount of money in my pocket is
directly related to how many hours I
work. The number of cheese sandwiches
I can afford is inversely related to how
much a cheese sandwich costs.
– What do I mean by this? What’s the
difference?
– What do the general forms of direct and
inverse variation equations look like?
Direct and Inverse Variation
Direct Variation
A simple linear relationship between two
variables where the two increase (or
decrease) together.
– Constant rate of increase.
– When one variable equals zero, the other one
does as well.
Direct Variation
# of bricks vs. height of stack
# of hours vs. amount of paycheck
# of minutes vs. distance from starting line
# of years old vs. # of days old
# of candy bars you buy vs. total cost
Direct Variations
y=kx
1. A constant rate of change.
2. When x is zero, y is also zero.
Direct Variation
Another way to look at things…
k is called the constant of variation.
– if y/x is not constant, it’s not a direct variation.
Direct Variation Skills
Is it a direct variation?
x
y
0
0
2
6
4
12
6
18
8
26
10
30
Direct Variation Skills
Is it a direct variation?
x
y
-3
15
-1
5
0
0
2
-10
4
-20
6
-30
Direct Variation Skills
What’s the constant of variation?
(12, 4)
Direct Variation Practice
Find the mising value in
the DV
(4, 32)
(x, 56)
What is the constant of
variation?
(4, 22)
Is this a direct variation?
If not, change the table
so it is, and write an
equation.
x
y
-1
-6
2
12
3
16
5
30
11
66
Inverse Variation
Mrs. Woodford is having a pizza party for
good referrals. Suppose there are 16
pieces in a pizza.
How many pieces will each student get
if….
– …only 1 student shows up?
– …2 students show up?
– …4 students show up?
– …10 students show up?
– …1,000,000 students show up?
Inverse Variation
As x gets farther from zero, y gets closer to zero.
There are two ways to talk about these:
In a direct variation, the ratio of x and y is constant.
In an indirect variation, the product of x and y is
constant.
For Example
Suppose the number of absences you have in a
semester relates to your grade in the following
way:
1 absence
8 absences
100 %
12.5 %
We test whether or not this is a indirect variation
by asking whether xy is constant.
If so, this is the constant of variation.
Inverse Variation
So a direct variation is sort of an upside
down direct variation:
– There are two quantities in an indirect
variation that react to one another in a very
predictable manner.
– There is a constant of variation that describes
this relationship.
– The difference is simply that in an indirect
variation, an increase in one quantity causes
a decrease in the other.
Inverse Variation Skills
Identifying IV’s, and
writing equations
x
y
2
18
4
9
12
3
Inverse Variation Skills
Identifying IV’s, and
writing equations
Cost
# of Items
1
100
10
10
20
5
Inverse Variation Skills
Finding the missing value for an IV.
(5, 8)
(80, x)
1 minute practice
Ms. Roberts teaches yoga in the
mornings, and has some mats.
– If 9 students were to show up, they’d each be
able to use two mats.
– If 36 students showed up, they’d each only
get half a mat.
1 minute practice
We won the jackpot! Wait – how many
people were in our lottery pool?
– If 2 of us share the jackpot, we each get
$250.
– If 40 of us share the jackpot, we each get
$12.50.
Inverse Variation Problems
They tend to be about sharing something.
This is a good way to recognize them in
word problems.
– Generally, there’s a limited amount of
something – pizza, cake, time, money,
whatever – to be split among people.
Practice
Identify which of these
are inverse variations and
which are direct, and
write the equation that
describes the
relationship.
x
y
4
18
6
27
10
45
x
y
x
y
x
y
3
10
-4
8
2
21
5
6
2
-4
4
10.5
2
15
7
-14
6
7
Practice
Find the missing values in these indirect
variations.
(4, 4)
(x, 8)
(20, ½ )
(2, x)
Summary:
Direct Variation
Another way to look at things…
k is called the constant of variation.
– if y/x is not constant, it’s not a direct variation.
Summary
As x gets farther from zero, y gets closer to zero (and vice
versa).
There are two ways to talk about these:
In an inverse variation, the product of x and y is
constant.