2.5 Proportion

Learning objectives

2.5HA Direct and inverse proportionset up direct or inverse proportion problems and relate algebraic solutions to graphs, using y ∝ x, y ∝ 1/x, y ∝ x², y ∝ 1/x², y ∝ x³, y ∝ 1/x³, y ∝ √x and y ∝ 1/√x

A proportionality statement specifies the shape of a relationship but not its scale. Replace $\propto$ by $=$ and a constant $k$ , then use known values to determine $k$ .

Statement	Equation	Graph behaviour for positive inputs
$y\propto x^n$	$y=kx^n$	passes through the origin for $n>0$
$y\propto1/x^n$	$y=k/x^n$	approaches the axes but is undefined at $x=0$
$y\propto\sqrt{x}$	$y=k\sqrt{x}$	increasing with decreasing gradient when $k>0$
$y\propto1/\sqrt{x}$	$y=k/\sqrt{x}$	decreasing for $x>0$ when $k>0$

Here $n$ is restricted to $1$ , $2$ or $3$ . The permitted forms are $x$ , $1/x$ , $x^2$ , $1/x^2$ , $x^3$ , $1/x^3$ , $\sqrt{x}$ and $1/\sqrt{x}$ .

If $y$ is inversely proportional to $x^2$ and $y=9$ when $x=2$ , write $y=k/x^2$ . Then $9=k/4$ , so $k=36$ and $y=36/x^2$ .

To find an input from an output, substitute into the completed equation, isolate the relevant power, then take the correct root. Use any domain condition to choose an allowed root.

A graph must match the algebraic form and constant: direct powers through the origin differ in curvature, while inverse powers have an excluded $x=0$ and branches shaped by parity and sign.

Never replace proportionality by equality without $k$ . Inverse proportion means reciprocal dependence such as $k/x^2$ , not merely a negative coefficient such as $-kx^2$ .

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—2.6 Simultaneous linear equationsRead