Direct Variation
Direct variation refers to a proportional relationship of two quantities.
e.g. the relationship between wage and hours worked
If one quanitity is altered in a certain ratio, the other quantity is affected the same way.
Notation for Proportional Variation
, varies directly with , describes:
Example
therefore
If and
Meaning the law connecting and for this particular velocity is
Graphs
Since the equation is equal to that of a straight line passing through the origin , any direct variation can be graphed accordingly.
Constant and Partly Variation
In practical work a variable might be dependent on a variable and a constant.
Where the values of and have to be figured out before the law can be stated.
Squared Direct Variation
,
Implying a quadratic function drawing a parabola when graphed.
For any kind of exponential variation the index is part of the law.
Cubed Direct Variation
,
The graph of is called a cubic curve.
Inverse Squared Variation
,
The graph only draws because a function may only yield one result.
Inverse Variation
,
As one value increases the other decreases or vice-versa.
Inverse variants when plotted appear as hyperbolas.
Joint Variation
When a quantity directly depends on two or more other quantities.
,
,
Determination of Non-Linear Laws
Laws can not be derived from exponential equations in a straightforward manner because their curves not not be derived in their entirety from their parts.
Example
Since plotting for would result in a parabola, plotting against and instead yields a straight line.
Logarithms can help finding unknown exponents:
So given two points and :
Subtracting the equations cancels :
Thus
Substituting in one of the equations now leads to the value of .