WriteWrite

The amount of money in my pocket is The amount of money in my pocket is directly relateddirectly related to how many hours I to how many hours I work. The number of cheese sandwiches work. The number of cheese sandwiches I can afford is I can afford is inversely relatedinversely related to how to how much a cheese sandwich costs.much a cheese sandwich costs.– What do I mean by this? What’s the What do I mean by this? What’s the

difference?difference?– What do the general forms of direct and What do the general forms of direct and

inverse variation equations look like?inverse variation equations look like?

Direct and Inverse VariationDirect and Inverse Variation

Direct VariationDirect Variation

A simple linear relationship between two A simple linear relationship between two variables where the two increase (or variables where the two increase (or decrease) together.decrease) together.

– Constant rate of increase.Constant rate of increase.– When one variable equals zero, the other one When one variable equals zero, the other one

does as well.does as well.

Direct VariationDirect Variation

# of bricks vs. height of stack# of bricks vs. height of stack

# of hours vs. amount of paycheck# of hours vs. amount of paycheck

# of minutes vs. distance from starting line# of minutes vs. distance from starting line

# of years old vs. # of days old# of years old vs. # of days old

# of candy bars you buy vs. total cost# of candy bars you buy vs. total cost

Direct VariationsDirect Variations

y=kxy=kx1.1. A constant rate of change.A constant rate of change.2.2. When x is zero, y is also zero.When x is zero, y is also zero.

– That is, there is That is, there is no starting conditionno starting condition..– This is the same as saying that the ratio of y/x is constant.This is the same as saying that the ratio of y/x is constant.

Contrary to popular belief, k Contrary to popular belief, k can can be negative. That is, be negative. That is, it’s not true that as x increases, y increases.it’s not true that as x increases, y increases.

Direct VariationDirect Variation

Another way to look at things…Another way to look at things…

kk is called the constant of variation. is called the constant of variation.– if y/x is if y/x is notnot constant, it’s not a direct variation. constant, it’s not a direct variation.

Direct Variation SkillsDirect Variation Skills

Is it a direct variation?Is it a direct variation? x y

0 0

2 6

4 12

6 18

8 26

10 30

Direct Variation SkillsDirect Variation Skills

Is it a direct variation?Is it a direct variation? x y

-3 15

-1 5

0 0

2 -10

4 -20

6 -30

Direct Variation SkillsDirect Variation Skills

What’s the constant of variation?What’s the constant of variation?

(12, 4)(12, 4)

Direct Variation PracticeDirect Variation Practice

Find the mising value in Find the mising value in the DVthe DV

(4, 32)(4, 32)(x, 56)(x, 56)

What is the constant of What is the constant of variation?variation?

(4, 22)(4, 22)

Is this a direct variation? Is this a direct variation? If not, change the table If not, change the table so it is, and write an so it is, and write an equation.equation.

x y

-1 -6

2 12

3 16

5 30

11 66

Inverse VariationInverse Variation

Mr. Bowen is having a stupid pizza party for Mr. Bowen is having a stupid pizza party for jerks. Suppose there are 16 pieces in a pizza.jerks. Suppose there are 16 pieces in a pizza.

How many pieces will each student get if….How many pieces will each student get if….– ……only 1 student shows up?only 1 student shows up?– ……2 students show up?2 students show up?– ……4 students show up?4 students show up?– ……10 students show up?10 students show up?– ……1,000,000 students show up?1,000,000 students show up?

Inverse VariationInverse Variation

As x gets As x gets farther from zerofarther from zero, y gets , y gets closer to zero.closer to zero.

There are two ways to talk about these:There are two ways to talk about these:

In a direct variation, the In a direct variation, the ratioratio of x and y is constant. of x and y is constant.In an indirect variation, the In an indirect variation, the productproduct of x and y is of x and y is constant.constant.

For ExampleFor Example

Suppose the number of absences you have in a Suppose the number of absences you have in a semester relates to your grade in the following semester relates to your grade in the following way:way:

1 absence1 absence 100 %100 %8 absences8 absences 12.5 % 12.5 %

We test whether or not this is a indirect variation We test whether or not this is a indirect variation by asking whether xy is constant.by asking whether xy is constant.If so, If so, this is the constant of variation.this is the constant of variation.

Inverse VariationInverse Variation

So a direct variation is sort of an upside So a direct variation is sort of an upside down direct variation:down direct variation:– There are two quantities in an indirect variation There are two quantities in an indirect variation

that react to one another in a very predictable that react to one another in a very predictable manner.manner.

– There is a There is a constant of variationconstant of variation that describes that describes this relationship.this relationship.

– The The difference difference is simply that is simply that in an indirect in an indirect variationvariation, an increase in one quantity causes a , an increase in one quantity causes a decrease in the other.decrease in the other.

Inverse Variation SkillsInverse Variation Skills

Identifying IV’s, and Identifying IV’s, and writing equationswriting equations

xx yy

22 1818

44 99

1212 33

Inverse Variation SkillsInverse Variation Skills

Identifying IV’s, and Identifying IV’s, and writing equationswriting equations

CostCost # of Items# of Items

11 100100

1010 1010

2020 55

Inverse Variation SkillsInverse Variation Skills

Finding the missing value for an IV.Finding the missing value for an IV.

(5, 8)(5, 8)

(80, x)(80, x)

1 minute practice1 minute practice

Ms. Roberts teaches yoga in the Ms. Roberts teaches yoga in the mornings, and has some mats.mornings, and has some mats.

– If 9 students were to show up, they’d each be If 9 students were to show up, they’d each be able to use two mats.able to use two mats.

– If 36 students showed up, they’d each only If 36 students showed up, they’d each only get half a mat.get half a mat.

1 minute practice1 minute practice

We won the jackpot! Wait – how many We won the jackpot! Wait – how many people were in our lottery pool?people were in our lottery pool?

– If 2 of us share the jackpot, we each get If 2 of us share the jackpot, we each get $250.$250.

– If 40 of us share the jackpot, we each get If 40 of us share the jackpot, we each get $12.50.$12.50.

Inverse Variation ProblemsInverse Variation Problems

They tend to be about They tend to be about sharingsharing something. something.

This is a good way to recognize them in This is a good way to recognize them in word problems.word problems.– Generally, there’s a limited amount of Generally, there’s a limited amount of

something – pizza, cake, time, money, something – pizza, cake, time, money, whatever – to be split among people.whatever – to be split among people.

PracticePractice

Identify which of these Identify which of these are inverse variations and are inverse variations and which are direct, and which are direct, and write the equation that write the equation that describes the describes the relationship.relationship.

x y

4 18

6 27

10 45

x y

3 10

5 6

2 15

x y

-4 8

2 -4

7 -14

x y

2 21

4 10.5

6 7

PracticePractice

Find the missing values in these Find the missing values in these indirectindirect variations.variations.

(4, 4)(4, 4) (20, ½ )(20, ½ )

(x, 8)(x, 8) (2, x) (2, x)

Summary: Summary: Direct VariationDirect Variation

Another way to look at things…Another way to look at things…

kk is called the constant of variation. is called the constant of variation.– if y/x is if y/x is notnot constant, it’s not a direct variation. constant, it’s not a direct variation.

SummarySummary

As x gets As x gets farther from zerofarther from zero, y gets , y gets closer to zero (and vice closer to zero (and vice versa).versa).

There are two ways to talk about these:There are two ways to talk about these:

In an inverse variation, the In an inverse variation, the productproduct of x and y is of x and y is constant.constant.