Variation+(Direct+and+Inverse)

= Aaron Floyd = = Direct and Inverse Variation = = Algebra II = =__Direct Variation__= When two variable quantities have a constant (unchanged) ratio, their relationship is called a direct variation. It is said that one variable "varies directly" as the other. The constant ratio is called the constant of variation. The formula for direct variation is y = kx, where k is the constant of variation. "y varies directly as x" Solving for k:   (y = numerator; x = denominator) Example: The weekly salary a woman earns, S, varies directly as the number of hours, h, which she works. Express this relation as a formula. Answer:    //S = hk//   or  (where k is the constant) CITATION : http://www.regentsprep.org/regents/math/variation/Ldirect.htm

Direct Variation hyperlink video: http://www.youtube.com/watch?v=wKXXrBV_RcA

Quiz Questions for Direct Variation: http://www.algebralab.com/practice/practice.aspx?file=Algebra_DirectVariation.xml

The statement " //y// varies directly as //x// ," means that when //x// increases, //y// increases //by the same factor.// In other words, //y// and //x// always have the same ratio:


 * [[image:http://img.sparknotes.com/figures/5/524e69123f5f88bd7240f5ec803934ef/latex_img2.gif]] = //k// ||  ||

where //k// is the constant of variation. We can also express the relationship between //x// and //y// as:


 * //y// = //kx// ||  ||

where //k// is the constant of variation. Since //k// is constant (the same for every point), we can find //k// when given any point by dividing the y-coordinate by the x-coordinate. For example, if //y// varies directly as //x//, and //y// = 6 when //x// = 2 , the constant of variation is //k// = = 3. Thus, the equation describing this direct variation is //y// = 3//x//. //Example 1// : If //y// varies directly as //x//, and //x// = 12 when //y// = 9 , what is the equation that describes this direct variation?

//k// = = //y// = //x// //Example 2// : If //y// varies directly as //x//, and the constant of variation is //k// = , what is //y// when //x// = 9 ?

//y// = //x// = (9) = 15 As previously stated, //k// is constant for every point; i.e., the ratio between the //y// -coordinate of a point and the //x// -coordinate of a point is constant. Thus, given any two points (//x//1, //y//1) and (//x//2, //y//2) that satisfy the equation, = //k// and  = //k//. Consequently, =  for any two points that satisfy the equation. //Example 3// : If //y// varies directly as //x//, and //y// = 15 when //x// = 10 , then what is //y// when //x// = 6 ?

=   =   6 = //y// //y// = 9

Graphing Direct Variation
An equation of the form //y// = //kx// can be thought of as an equation of the form //y// = //mx// + //b// where //m// = //k// and //b// = 0. Thus, a direct variation equation is an equation in slope- intercept form which passes through (0, 0) and has a slope equal to the constant of variation. Therefore, to graph a direct variation equation, start at (0, 0) and then proceed as you would in graphing a slope. Or, if you know one point, draw a straight line between (0, 0) and that point, and extend the line on both sides. //Example 4// : //y// varies directly as //x//. If the constant of variation is, graph the line which represents the variation, and write an equation that describes the variation. y //= //x To calculate the constant of variation, given a graph of direct variation, simply calculate the slope. Citation: http://www.sparknotes.com/math/algebra1/variation/section1.html

__Link to Practice Problems for Direct Variation__: http://www.westirondequoit.org/ihs/Math/mathamastery/direct_variation.htm Citation: __Direct Variation__. 14 Nov. 2008 .

(The Opposite of Direct Variation)**
 * **__Inverse Variation__

In an ** inverse variation **, the values of the two variables change in an opposite manner - as one value increases, the other decreases.

For instance, a biker traveling at 8 mph can cover 8 miles in 1 hour. If the biker's //speed decreases// to 4 mph, it will take the biker 2 hours (//an increase of one hour)//, to cover the same distance. Inverse variation: when one variable //increases//, the other variable //decreases.// || As speed decreases, the time increases. || Notice the shape of the graph of inverse variation. If the value of //x// is increased, then //y// decreases. If //x// decreases, the //y// value increases. We say that **//y// varies inversely as the value of //x//**. An ** inverse variation ** between 2 variables, //y// and //x//, is a relationship that is expressed as: where the variable //k// is called the // constant of proportionality.

// Citation: http://www.regentsprep.org/Regents/math/algtrig/ATE7/Inverse%20Variation.htm

Inverse Variation Hyperlink video: http://www.youtube.com/watch?v=YkGBiyZgElM