Direct and inverse proportion

​​​​In a nutshell

Proportion describes a relationship between two sets of quantities which are related by the constant ratio, $$k$$. The symbol $$\propto$$ means "is proportional to".​

Direct proportion

Two directly proportional quantities are related by a constant $$k$$ such that if $$y \propto x^n$$ then $$y = kx^n$$. This means, as $$y$$ increases, $$x^n$$ increases whilst maintaining a constant ratio of $$k$$ between their quantities.​​

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Note: Linear models show proportion where $$(y-c) \propto x$$​ with the constant ratio being the gradient, $$m$$.​

Example 1

When $$y= 45$$, $$x=3$$. Given that $$y$$ is proportional to $$x^2$$, find the value of $$y$$​ when $$x=2$$.​

Represent the relationship of $$y$$ and $$x$$ as an equation.​

$$\begin{aligned} y &\propto x^2 \\ y&=kx^2 \end{aligned}$$​

Find the value of the constant $$k$$ by inserting the known values.

$$\begin{aligned}(45) &= k(3)^2 \\ 45 &=9 k \\ 5&=k \end{aligned}$$

To find $$y$$ when $$x=2$$, substitute $$x=2$$ into $$y=kx^2$$, where $$k=5$$.​

$$\begin{aligned}y&= (5)(2)^2 \\ y&=(5)(4) \\ y &= 20 \end{aligned}$$​​

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Therefore, when $$\underline{ x =2, \ y=20}$$.

Inverse proportion

Two inversely proportional quantities are also related by the constant ratio, $$k$$ , but such that if $$y \propto \dfrac 1{x^n}$$​ then $$y = \dfrac{k}{x^n}$$​. This means as $$y$$​ increases, $$x^n$$​ decreases and vice versa whilst maintaining a constant ratio of $$k$$​ between their quantities.​​

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Example 2

When $$y= 1$$, $$x=1$$. Given that $$y$$ is inversely proportional to $$x^2+1$$, find the value of $$y$$​ when $$x=3$$.​

Represent the relationship of $$y$$ and $$x$$ as an equation.​

$$\begin{aligned} y &\propto \dfrac{1}{x^2 +1} \\ \\ y &= \dfrac{k}{x^2 +1}\end{aligned}$$​

Find the value of the constant $$k$$.

$$\begin{aligned}(1) &= \dfrac {k}{(1)^2+1} \\ \\ 1 &=\dfrac {k}{2} \\ \\ 2&=k \end{aligned}$$

Substitute $$x=3$$ into the equation $$y = \dfrac{2}{x^2 +1}$$.

$$\begin{aligned}y&= \dfrac{2}{(3)^2+1} \\ \\ y&= \dfrac{2}{10}\\ \\ y &= \dfrac15 = 0.2 \end{aligned}$$​​

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Therefore, when $$\underline{ x =3, \ y=0.2}$$.

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Sketching graphs of proportion

After relating two proportional quantities by a constant $$k$$, it is possible to form an equation in order to sketch a graph .​

procedure

Note: Graphs of variables which are inversely proportional will often produce asymptotes. An asymptote is a line that gets closer and closer to a curve but never touches it.

Example 3

Given that $$y$$ is inversely proportional to $$x+3$$, when $$y=2$$, $$x=7$$. Sketch a graph to represent this relationship.​

Create an equation to relate the quantities. 

$$\begin{aligned} y &\propto \dfrac{1}{x+3} \\ \\ y &= \dfrac{k}{x+3} \end{aligned}$$​​

Solve for constant $$k$$.

$$\begin{aligned} (2) &= \dfrac {k}{(7)+3} \\ \\ 2&= \dfrac {k}{10} \\ \\ 20 &=k \end{aligned}$$​​

Write an equation in terms of $$x$$ and $$y$$.

$$y = \dfrac{20}{x+3}$$​​

Find any asymptotes. Vertical asymptotes occur when the denominator here equals zero.

$$\begin{aligned}0 &= x+3 \\ x&=-3\\ \\ y&=0 \end{aligned}$$​​

Find the $$y\text-intercept$$ by setting $$x=0$$.

$$y = \dfrac{20}{(0)+3} = \dfrac{20}3$$​​

Note: In some cases there isn't a $$y\text-intercept$$. In this case there will be no value for $$y$$ when $$x=0$$. Likely this will mean there is an asymptote on the $$y$$ axis.

Use the information gathered to sketch the curve.