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Direct Variation

A relationship between two variables in which one variable is a constant multiple of the other. When graphing the line DOES pass through the origin.

Represented by y=mx form

X and Y values vary directly with each other

Partial Variation

A relationship between two variables in which one variable is a constant multiple of the other plus a constant value. Graph DOES NOT pass through the origin.

Represented by y=mx+b form

X and Y values don't vary directly with each other

The Equation

y=mx and y=mx+b

Y is the unknown value

M is the slope or how much the line increases

B is the Y-intercept

X is the value on the X-axis

Equation Direct Variation:
y=mx
M: is the constant of variation

Partial Variation:
y=mx+b
M: is the constant of variation
B: is the fixed cost

Characteristics

Direct Variation

Partial Variation

straight line

straight line

constant of variation

constant of variation

no fixed cost

fixed cost

x and y values

x and y values

starts at origin (0,0)

starts anywhere but origin

y=mx

y=mx+b

Diagrams (graph format)

Partial Variations

Direct Variations

Direct Variation Examples

Example #1

Papoose the cat earns 2 cat treats an hour for sleeping. How many cat treats does he earn for the first 6 hours?

y=2x

Hours

Cat Treats Earned

0

0

1

2

2

4

3

6

4

8

5

10

6

12

In this case, 2 is the cat treats earned (m) and x is the number of hours Papoose was sleeping. The cat treats earned is represented by y.

If this equation was graphed it would start at the origin (0,0), and go up by 2 each time.

Example #2

Remember y=mx

Savard earns a wage of $9.00/hour. (She's been working really hard and shown dedication as a janitor so she got a massive raise, from minimum wage). This table shows the money that Savard earns in the first 5 hours of work.

Hours

Money Earned ($)

0

0

1

9

2

18

3

27

4

36

5

45

Notice at 0 hours $0 are made. Savard's table displays the relation between her hours worked (as a Janitor) and her wages. Her wages vary directly with the hours worked, this relation is a direct variation.

y=9x
The "y" is her wages.
The "x" is her hours worked.
Y varies directly with x.

*This is a real world example*

Example #3

John is driving his Caravan down the highway at 90 km/hour. Complete the table of values to find out how many kilometers John travels after 5 hours.

Number of Hours(x)

Equation

KM per Hour (y)

0

y=90(0)

0

1

y=90(1)

90

2

y=90(2)

180

3

y=90(3)

270

4

y=90(4)

360

5

y=90(5)

450

Therefore, John travels 450 km after 5 hours. This is direct variation

Example #4

Steve Yzerman gets one medal every-time he visits a city. This table represents his relationship between the number of cities and number of medals.

Number of Medals

Number of Cities

0

0

1

1

2

2

3

3

4

4

As you can clearly see the relationship is direct because the number of medals varies directly with the number of cities. The equation that represents this is: y=1x, where x is the number of medals and y is the number of cities.

Example #5

Doug Gilmour scores 3 hat-tricks a month.
The table below shows the relationship between the number of months and the number of hat-tricks.

Number of Months (X)

Number of Hat-Tricks (Y)

0

0

1

3

2

6

3

9

4

12

As you can see each month he gets 3 hat-tricks, this is direct variation and the equation is y=3x.

Example #6 Mr. Datsyuk and Mrs. Zetterburg are riding there two seated Barbie Scooter down the highway.
The formula that describes their motion is y = 80x.
Complete the table of values and graph the relation.

X Number of Hours

Y KM Traveled

0

0

2

160

4

320

6

480

8

640

10

800

12

960

Here are some videos that may help some people:

Partial Variation Examples

Papoose the Cat gets a babysitter to watch over him while his owner is at work. The babysitter loves to spoil cats and gives Papoose a daily 2 treats and adds on another 3 treats per hour as well.

y=3x+2

Hours

Cat Treats Earned

0

2

1

5

2

8

3

11

4

14

5

17

6

20

In this case, +2 is the number of treats (b) Papoose gets for just doing, well, nothing. His babysitter gives him 2 just for playing. Anyways, 2 is the initial value (b) where the graph will start. 3 is the number that shows how many cat treats Papoose earns every hour. "x" represents the amount of hours. "y" represents the total amount of cat treats earned every hour, including the daily 2 cat treats.

If this was graphed, it would start at the 2 on the y-axis and constantly go up by 3.

Example 2:

A car repair shop charges $45/hour plus a garage fee of $60. Complete the table of values.

Hours

Cost of Repair

0

60

1

105

2

150

3

195

4

240

5

285

6

330

As you can see there is a automatic rate of $60. You also have to add the hourly cost of $45. This is why even at 0 hours there is still a cost. You then continue to add 45 each time. The equation for this example is y=45x+60. The 45 is the cost of variation, the 60 is the fixed cost. Y represents the cost of the repair and X represents the number of hours worked. This is partial variation.

*This is a real world example*

Example #3

Roy Halladay earns $25 for every hour that he pitches, as well as a start-up pay of $40.
Complete the table of values.

Hours(X)

Salary (Y)

0

$40

1

$65

2

$90

3

$115

4

$140

5

$165

6

$190

7

$215

Even though Roy has not worked when the hours are 0, he still starts off with $40. This is called the fixed cost. He gains $25 for every hour, as well as the $40 that he starts with. Therefore, the equation for Roy's earnings is y=25x+40, where x is the number of hours and y is his total earnings. This is partial variation

Example #4

The Kool-Aid man was selling his spectacular juice at a store. He found that there was $30 in his cash register.

He was selling bottles of juice for $1.50/bottle. Complete the table of values to see how much money he had, in the cash register, after he sold the juice bottles to 5 customers.

Customers (X)

Equation

Money (Y)

0

y=1.50(0)+30

$30

1

y=1.50(1)+30

$31.50

2

y=1.50(2)+30

$33

3

y=1.50(3)+30

$34.50

4

y=1.50(4)+30

$36

5

y=1.50(5)+30

$37.50

Therefore, he had $37.50 after 5 customers, and this is partial variation. The equation is y=1.5x+30.

Example #5

Imadehimup Johnson broke his arm while riding his bike. For each day that he stayed,at the hospital, he needed to pay $25. To get into the parking lot, his mom had to a pay $10 entry fee once.

Complete the table of values to see how much the hospital bill was after 5 nights at the hospital.

of Days at the Hospital (X)

Hospital Bill (Y)

0

$10

1

$35

2

$60

3

$85

4

$110

5

$135

Therefore, his hospital bill after 5 days was $135. This is partial variation. Graphing Equations

Direct Variation:

When graphing Direct Variation you must use the equation y=mx to find your "x" values and "y" values. The "x" value will represent the co-ordinate on the x-axis and the "y" value will represent the co-ordinate on the y-axis. In Direct Variation the Y-Intercept MUST be at the origin. Also the "m" is actually the slope but that ties in with analytical geometry.

Partial Variation:

When graphing Partial Variation instead of the equation y=mx, you will use y=mx+b to find the "x" and "y" values. Like Direct Variation, the "x" value will represent the co-ordinate on the x-axis and the "y" value will represent the co-ordinate on the y-axis. In the Partial Variation the Y-Intercept is the "b" value.

Other Facts

Partial variation also ties in with analytical geometry! Since the equation for partial variation is y=mx+b, it is also used to make the equation of a linear line. Although, in the equation of a linear line 'm' represents the slope and the 'b' represents the y-intercept! "Y" and "X" are still the y and x co-ordinates on the graph though.

Direct vs. Partial Variation!

(MARKED 2009/06/07)See the bottom of our Wiki for Podcasts, Powerpoints, Excel graphs and other things!

Direct Variation

Partial VariationThe Equation

EquationDirect Variation:y=mx

M: is the constant of variation

Partial Variation:y=mx+b

M: is the constant of variation

B: is the fixed cost

## Characteristics

Direct VariationPartial Variation## Diagrams (graph format)

Partial VariationsDirect VariationsDirect Variation ExamplesExample #1

Papoose the cat earns 2 cat treats an hour for sleeping. How many cat treats does he earn for the first 6 hours?

y=2xIn this case, 2 is the cat treats earned (m) and x is the number of hours Papoose was sleeping. The cat treats earned is represented by y.

If this equation was graphed it would start at the origin (0,0), and go up by 2 each time.

Example #2

Remember

y=mxSavard earns a wage of $9.00/hour. (She's been working really hard and shown dedication as a janitor so she got a massive raise, from minimum wage). This table shows the money that Savard earns in the first 5 hours of work.

y=9x

The "y" is her wages.

The "x" is her hours worked.

Y varies directly with x.

*This is a real world example*

## Example #3

John is driving his Caravan down the highway at 90 km/hour. Complete the table of values to find out how many kilometers John travels after 5 hours.

Example #4Steve Yzerman gets one medal every-time he visits a city. This table represents his relationship between the number of cities and number of medals.

As you can clearly see the relationship is direct because the number of medals varies directly with the number of cities. The equation that represents this is: y=1x, where x is the number of medals and y is the number of cities.

Example #5Doug Gilmour scores 3 hat-tricks a month.

The table below shows the relationship between the number of months and the number of hat-tricks.

As you can see each month he gets 3 hat-tricks, this is direct variation and the equation is y=3x.

Example #6Mr. Datsyuk and Mrs. Zetterburg are riding there two seated Barbie Scooter down the highway.

The formula that describes their motion is y = 80x.

Complete the table of values and graph the relation.

Here are some videos that may help some people:

Partial Variation ExamplesPapoose the Cat gets a babysitter to watch over him while his owner is at work. The babysitter loves to spoil cats and gives Papoose a daily 2 treats and adds on another 3 treats per hour as well.

y=3x+2If this was graphed, it would start at the 2 on the y-axis and constantly go up by 3.

Example 2:A car repair shop charges $45/hour plus a garage fee of $60. Complete the table of values.

HoursCost of Repair*This is a real world example*

Example #3Roy Halladay earns $25 for every hour that he pitches, as well as a start-up pay of $40.

Complete the table of values.

fixed cost.He gains $25 for every hour, as well as the $40 that he starts with. Therefore, the equation for Roy's earnings is y=25x+40, where x is the number of hours and y is his total earnings. This is partial variation## Example #4

The Kool-Aid man was selling his spectacular juice at a store. He found that there was $30 in his cash register.He was selling bottles of juice for $1.50/bottle. Complete the table of values to see how much money he had, in the cash register, after he sold the juice bottles to 5 customers.

Customers (X)EquationMoney (Y)Therefore, he had $37.50 after 5 customers, and this is partial variation. The equation is y=1.5x+30.

Imadehimup Johnson broke his arm while riding his bike. For each day that he stayed,at the hospital, he needed to pay $25. To get into the parking lot, his mom had to a pay $10 entry fee once.Example #5Complete the table of values to see how much the hospital bill was after 5 nights at the hospital.

Therefore, his hospital bill after 5 days was $135. This is partial variation.

Graphing Equations

Direct Variation:When graphing Direct Variation you must use the equation y=mx to find your "x" values and "y" values. The "x" value will represent the co-ordinate on the x-axis and the "y" value will represent the co-ordinate on the y-axis. In Direct Variation the Y-Intercept MUST be at the origin. Also the "m" is actually the slope but that ties in with analytical geometry.

Partial Variation:When graphing Partial Variation instead of the equation y=mx, you will use y=mx+b to find the "x" and "y" values. Like Direct Variation, the "x" value will represent the co-ordinate on the x-axis and the "y" value will represent the co-ordinate on the y-axis. In the Partial Variation the Y-Intercept is the "b" value.

Other Facts

Here are two Podcast casts I made, enjoy!