Direct and inverse variation

Direct and inverse variation (proportion), finding the constant of variation, variation with powers and square roots, and real-world problems.

Direct variation

"$y$ varies directly with $x$" (or "$y$ is proportional to $x$") means $y = kx$ for some constant $k$. If $x$ doubles, $y$ doubles. The graph is a straight line through the origin.

Direct variation

$$y \propto x \quad\Longleftrightarrow\quad y = kx$$

Inverse variation

"$y$ varies inversely with $x$" means $y = \dfrac{k}{x}$. If $x$ doubles, $y$ halves. The graph is a hyperbola.

Inverse variation

$$y \propto \frac{1}{x} \quad\Longleftrightarrow\quad y = \frac{k}{x}$$

Variation with powers

n = 1: direct. n = 2: with the square. n = 0.5: with the square root. n = −1: inverse.

Try this

Set $n = 2$. Check that doubling $x$ multiplies $y$ by $4$.
Set $n = -1$. What happens to $y$ when $x$ doubles?

The four-step method

Example 1●

The distance $d$ a car travels at constant speed varies directly with time $t$. It travels $150$ km in $2$ hours. How far does it travel in $5$ hours?

$d = kt$

$150 = 2k$, so $k = 75$.

Substitute d = 150 and t = 2.

$d = 75t$

When $t = 5$, $d = 375$ km.

Example 2●●

The intensity $I$ of light varies inversely with the square of the distance $d$ from the source. At $2$ m, $I = 90$ units. Find $I$ at $3$ m.

$I = \dfrac{k}{d^2}$

Inversely with the square means k divided by d².

$90 = \dfrac{k}{4}$, so $k = 360$.

$I = \dfrac{360}{d^2}$, so at $d = 3$: $I = \dfrac{360}{9} = 40$ units.

Example 3: what if it triples?●●●

The volume $V$ of a sphere varies directly with the cube of its radius. What happens to the volume when the radius is tripled?

$V = kr^3$

New volume $= k(3r)^3 = 27kr^3 = 27V$.

Replace r with 3r.

The volume is multiplied by $27$.

Flashcards

y varies directly with x

$y = kx$

y varies inversely with x

$y = \dfrac{k}{x}$

y varies with the square of x

$y = kx^2$. Doubling $x$ multiplies $y$ by $4$.

How to find k

Substitute one known pair of values.

Practice

$P$ varies directly with the square root of $Q$. Which equation fits?
$y$ varies inversely with $x$, and $y = 12$ when $x = 2$. Find $y$ when $x = 3$.
$A$ varies with the square of $r$. If $r$ is halved, $A$ is:
$y = kx^2$ and $y = 20$ when $x = 2$. Find $y$ when $x = 3$.
$y$ varies directly with $x$. If $x$ is tripled, $y$ is:
$y$ varies inversely with $x^2$, and $y = 4$ when $x = 3$. Find $k$.
Which graph shows direct variation?
$P$ varies directly with $\sqrt{Q}$, and $P = 12$ when $Q = 9$. Find $P$ when $Q = 25$.