Direct+and+Partial+Variations



Direct vs. Partial Variation! **(MARKED 2009/06/07)**

See the bottom of our Wiki for Podcasts, Powerpoints, Excel graphs and other things!

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**  y=mx M: is the <span style="font-size: 130%; color: rgb(216, 105, 147);">constant of variation <span style="font-size: 130%; color: rgb(0, 0, 0);">
 * Direct Variation:**

y=mx+b M: is the <span style="font-size: 130%; color: rgb(216, 105, 147);">constant of variation <span style="font-size: 130%; color: rgb(0, 0, 0);"> B: is the <span style="font-size: 130%; color: rgb(248, 180, 73);">fixed cost
 * Partial Variation:**

=<span style="color: rgb(248, 144, 48); font-family: 'Comic Sans MS',cursive;"> = =<span style="color: rgb(248, 144, 48); font-family: 'Comic Sans MS',cursive;"> = =<span style="color: rgb(3, 3, 3); font-family: 'Comic Sans MS',cursive;">Characteristics = <span style="color: rgb(248, 144, 48); font-family: 'Comic Sans MS',cursive;">


 * = **<span style="color: rgb(22, 243, 109);">Direct Variation ** ||= **<span style="color: rgb(22, 243, 109);">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 ||

<span style="color: rgb(8, 7, 7); font-family: 'Comic Sans MS',cursive;">Diagrams (graph format)

 * <span style="color: rgb(34, 240, 247); font-family: 'Comic Sans MS',cursive;">Partial Variations **

<span style="color: rgb(34, 240, 247); font-family: 'Comic Sans MS',cursive;">**Direct Variations** <span style="color: rgb(255, 79, 0);">

<span style="font-size: 190%; color: rgb(17, 208, 99); text-align: center; display: block;">
 * Direct Variation Examples**

<span style="font-size: 130%; color: rgb(244, 0, 255); font-family: 'Comic Sans MS',cursive;">Example #1 <span style="color: rgb(244, 0, 255); font-family: 'Comic Sans MS',cursive;"> <span style="display: block; font-size: 13px; color: rgb(244, 0, 255); font-weight: normal; line-height: 19px;"> Papoose the cat earns 2 cat treats an hour for sleeping. How many cat treats does he earn for the first 6 hours?

<span style="color: rgb(248, 48, 79);">** y=2x **
 * Hours || Cat Treats Earned ||
 * 0 || 0 ||
 * 1 || 2 ||
 * 2 || 4 ||
 * 3 || 6 ||
 * 4 || 8 ||
 * 5 || 10 ||
 * 6 || 12<span style="font-size: 120%; color: rgb(255, 198, 0); font-family: 'Comic Sans MS',cursive;"> ||

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.

<span style="font-size: 120%; color: rgb(34, 240, 247); font-family: 'Arial Black',Gadget,sans-serif;">Example #2 <span style="font-size: 120%; color: rgb(255, 3, 0);">

Remember <span style="font-size: 120%; color: rgb(255, 3, 0); font-family: 'Comic Sans MS',cursive;">**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. Notice at 0 hours $0 are made. Savard's table displays the relation between her hours worked (as a Janitor) and her wages. Her wages vary directly with the hours worked, this relation is a direct variation.
 * < Hours ||< Money Earned ($) ||
 * < 0 ||< 0 ||
 * < 1 ||< 9 ||
 * < 2 ||< 18 ||
 * < 3 ||< 27 ||
 * < 4 ||< 36 ||
 * < 5 ||< 45 ||

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


 * This is a real world example*

=<span style="color: rgb(216, 105, 147);">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. Therefore, John travels 450 km after 5 hours. This is direct variation
 * < # 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 ||

<span style="font-size: 130%; color: rgb(58, 186, 248);">**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.

//<span style="color: rgb(255, 0, 35);">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.

<span style="display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;">Mr. Datsyuk and Mrs. Zetterburg are riding there two seated Barbie Scooter down the highway. <span style="display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;"> The formula that describes their motion is y = 80x. Complete the table of values and graph the relation.
 * <span style="font-size: 110%; color: rgb(34, 240, 247);">Example #6 **
 * X Number of Hours || Y KM Traveled ||
 * 0 || 0 ||
 * 2 || 160 ||
 * 4 || 320 ||
 * 6 || 480 ||
 * 8 || 640 ||
 * 10 || 800 ||
 * 12 || 960 ||

<span style="color: rgb(170, 170, 170);">

Here are some videos that may help some people: media type="youtube" key="jvHk48-Vw-M" height="344" width="425"media type="youtube" key="wKXXrBV_RcA" width="425" height="350"

=**<span style="display: block; font-size: 160%; color: rgb(255, 110, 0); font-family: 'Comic Sans MS',cursive; text-align: center;">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.


 * <span style="font-size: 140%; color: rgb(200, 237, 33); font-family: 'Comic Sans MS',cursive;">y=3x+2[[image:http://www.pantherkut.com/wp-content/uploads/2007/06/fat_cat_4.jpg width="193" height="168" align="right"]] **

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.
 * Hours || Cat Treats Earned ||
 * 0 || 2 ||
 * 1 || 5 ||
 * 2 || 8 ||
 * 3 || 11 ||
 * 4 || 14 ||
 * 5 || 17 ||
 * 6 || 20 ||

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

 <span style="background-color: rgb(255, 255, 255);">**<span style="font-size: 120%; color: rgb(235, 142, 225); text-decoration: none;">Example 2: **

 A car repair shop charges $45/hour plus a garage fee of $60. Complete the table of values. <span style="background-color: rgb(255, 255, 255);">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.
 * = __**Hours**__ ||= __**Cost of Repair**__ ||
 * = 0 ||= <span style="background-color: rgb(255, 255, 255);">60 ||
 * = 1 ||= <span style="background-color: rgb(255, 255, 255);">105 ||
 * = 2 ||= <span style="background-color: rgb(255, 255, 255);">150 ||
 * = 3 ||= <span style="background-color: rgb(255, 255, 255);">195 ||
 * = 4 ||= <span style="background-color: rgb(255, 255, 255);">240 ||
 * = 5 ||= <span style="background-color: rgb(255, 255, 255);">285 ||
 * = <span style="background-color: rgb(255, 255, 255);">6 ||= 330 ||


 * 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. 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
 * Hours(X) || Salary (Y) ||
 * 0 || $40 ||
 * 1 || $65 ||
 * 2 || $90 ||
 * 3 || $115 ||
 * 4 || $140 ||
 * 5 || $165 ||
 * 6 || $190 ||
 * 7 || $215 ||

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.

<span style="font-size: 180%; color: rgb(255, 3, 0); font-family: 'Comic Sans MS',cursive;">Graphing Equations

<span style="font-size: 120%; color: rgb(248, 98, 13);">**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.

<span style="font-size: 120%; color: rgb(220, 96, 159);">**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.

<span style="font-size: 196%; color: rgb(246, 185, 132); font-family: 'Comic Sans MS',cursive;"> **
 * <span style="font-size: 168%; color: rgb(107, 235, 162); font-family: 'Comic Sans MS',cursive;">Other Facts
 * Partial variation also ties in with <span style="color: rgb(248, 180, 73);">analytical geometry ! Since the equation for partial variation is <span style="color: rgb(248, 180, 73);">y=mx+b, it is also used to make the equation of a linear line. Although, in the equation of a linear line '<span style="color: rgb(248, 180, 73);">m ' represents the slope and the '<span style="color: rgb(248, 180, 73);">b ' represents the y-intercept! "Y" and "X" are still the y and x co-ordinates on the graph though.

Here are two Podcast casts I made, enjoy!

media type="file" key="Podcast 1.mov"media type="file" key="Podcast 2.mov"


 * [[file:Wiki math powerpoint.ppt]]**

Ryan's Direct & Partial Variation Powerpoint
==

=Brendan's Direct vs. Partial Variation Powerpoint=

[[file:Direct Variation Graph.xls]]
=Brendan's Direct Variation Graph=

[[file:wiki comic.comicdoc]]
=Brendan's Wiki Comic=

[[file:Partial Variation Graph.xls]]
=Brendan's Partial Variation Graph=

<span style="color: rgb(0, 0, 0);"> <span style="color: rgb(58, 186, 248);">
 * <span style="font-size: 140%; color: rgb(255, 0, 165); font-family: 'Comic Sans MS',cursive;">Citations **
 * Mah, R., (1993). //Direct and Partial Variations//. Retrieved December 17, 2008, from http://mathcentral.uregina.ca/RR/database/RR.09.96/mah1.html
 * <span style="color: rgb(0, 0, 0);">McGraw-Hill Ryerson Limited. (1999). //Mathpower 9 Ontario Edition//. Toronto: Diane Wyman.
 * //Math Variation: Direct and Indirect// (2007, September 15).
 * Rancourt, S., ( Wed. Apr. 8th, & Thurs. Apr. 9th, 2009)MPM 1D1- Course Notes. Retrieved June 3, 2009.
 * Rancourt, S. (2009). Direct Variation..
 * Rancourt, S. (2009). Partial Variation..
 * Rancourt, S. (2009). MPM 1D1 - Course Notes

<span style="color: rgb(212, 100, 247);">Page created by: <span style="color: rgb(255, 0, 208);"> Ryan Duffy, Brendan Holmes, and Mat D'Ortenzio