Direct Proportion with Powers and Roots

Extending direct proportion

In linear direct proportion, $y\propto x$ becomes $y=kx$. The same idea works when $y$ is directly proportional to a power or root of $x$: $y$ is a constant multiple of that power or root.

For example:

• $y\propto x^2$ means $y=kx^2$;
• $y\propto x^3$ means ;

Here, $k$ is the constant of proportionality. The symbol states the relationship; introducing turns it into an equation. It would therefore be incorrect to change straight into unless the evidence shows that .

What stays constant

For $y\propto x$, the ratio $y/x$ is constant. When a power or is involved, the constant ratio changes accordingly:

$$y\propto x^2 \quad\Longrightarrow\quad \frac{y}{x^2}=k$$ $$y\propto \sqrt{x} \quad\Longrightarrow\quad \frac{y}{\sqrt{x}}=k$$

This explains why the relevant expression must be included when calculating $k.$ If $y\propto x^3$, it is $y/x^3$—not $$—that remains constant.

The scaling also follows the stated power or root. If $y\propto x^2$ and $x$ is multiplied by $3$, then $y$ is multiplied by $3^2=9$. If and is multiplied by , then is multiplied by .

Forming and using the equation

Use the same four-stage method for any power or root relationship:

1. Translate the words into a formula containing $k$.
2. Substitute a known pair of corresponding values.
3. Solve for $k$ and write the complete equation.
4. Substitute the new value required by the question.

Suppose $y$ is directly proportional to the square of $x$, and $y=18$ when $x=3$. First write

$$y=kx^2.$$

Substituting the known pair gives

$$18=k(3^2)=9k,$$

so $k=2$. The equation connecting the variables is therefore

$$y=2x^2.$$

When $x=4$,

$$y=2(4^2)=32.$$

The value $32$ is not found by scaling $18$ in the ordinary ratio $4:3$. Because the relationship uses a square, the correct scale factor is $(4/3)^2$.

Working with roots

The method is unchanged when the relationship contains a root. Suppose $$ is directly proportional to the square root of , and when . Then

$$p=k\sqrt{q}.$$

Using the known values,

$$15=k\sqrt{25}=5k,$$

so $k=3$ and the complete equation is

$$p=3\sqrt{q}.$$

When $q=64$, this gives $p=3\sqrt{64}=24$.

An equation can also be used backwards. From $p=3\sqrt{q}$, if $p=21$, then $$, so and hence .

Recognising the graph

Only the basic relationship $y=kx$ necessarily gives a straight line through the origin. A square, cube or root usually produces a curve with a different shape. Nevertheless, the relationship is still because remains a constant multiple of the stated expression.

Data for Direct proportion to a square and a square root

The graph reinforces an important distinction: the square relationship has constant $y/x^2$, while the square-root relationship has constant $y/\sqrt{x}$. Neither generally has a constant value of .

Translate the statement

• $y\propto x^2 \Rightarrow y=kx^2$
• $$