XXIV Variation and Laws

Direct Variation

Direct variation refers to a proportional relationship of two quantities.

e.g. the relationship between wage and hours worked

$\frac {W$1} {W2} = \frac {T1} {T2}

$\frac {3200} {6400} = \frac {120} {240}$

If one quanitity is altered in a certain ratio, the other quantity is affected the same way.

Notation for Proportional Variation

$y \varpropto x$, $y$ varies directly with $x$, describes:

$y = kx$

Example

$s = \text{Distance travelled}$
$t = \text{Time}$

$s \varpropto t$
$s = kt$ therefore $k = \frac s t$

If $s = 40$ and $t=2.5$

$40=2.5k$
$k = \frac {40}{2.5}$
$k = 16$

Meaning the law connecting $s$ and $t$ for this particular velocity is

$s = 16t$

Graphs

Since the equation $y = kx$ is equal to that of a straight line passing through the origin $y = mx$, 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.

$y = ax - b$

Where the values of $a$ and $b$ have to be figured out before the law can be stated.

Squared Direct Variation

$y \varpropto x^2$, $y = kx^2$

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

$y \varpropto x^3$, $y = kx^3$

The graph of $x^3$ is called a cubic curve.

Inverse Squared Variation

$y \varpropto \sqrt[2]{x}$, $y = x^{\frac 1 2}$

The graph only draws $y = \sqrt[+]{x}$ because a function may only yield one result.

Inverse Variation

$y \varpropto \frac 1 x$, $y = \frac k x$

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.

$y \varpropto xz$, $y = kxz$

$y \varpropto \frac x z$, $y = \frac {kx}{z}$

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

$y = ax^n + b$

Since plotting for $x$ would result in a parabola, plotting against $x^n$ and $y$ instead yields a straight line.

Logarithms can help finding unknown exponents:
$y = cx^n$
$\log y = n \log x + \log c$

So given two points $(x$1, y1) and $(x$2, y2):

$3.236 = 1.398n + \log c$
$2.795 = 1.255n + \log c$

Subtracting the equations cancels $\log c$:

$0.441 = 0.143n$

Thus

$n = \frac {0.441}{0.143} \approx 3.1$

Substituting $n$ in one of the equations now leads to the value of $c$.