Direct and Inverse Variation

Direct Variation

$y = kx$

where $k$ is the constant of variation (or constant of proportionality).

• As $x$ increases, $y$ increases.
• The ratio $\frac{y}{x} = k$ is constant.
• The graph passes through the origin.

Example: If $y$ varies directly with $x$ and $y = 15$ when $x = 3$, find $k$ and the equation.

$k = \frac{15}{3} = 5$. Equation: $y = 5x$.

Inverse Variation

$y = \frac{k}{x} \quad \text{or equivalently} \quad xy = k$

where $k$ is the constant of variation.

• As $x$ increases, $y$ decreases.
• The product $xy = k$ is constant.
• The graph is a hyperbola.

Example: If $y$ varies inversely with $x$ and $y = 6$ when $x = 4$, find $k$.

$k = xy = 24$. Equation: $y = \frac{24}{x}$.

Identifying from Tables

Direct: Check if $\frac{y}{x}$ is constant.

$x$	$y$	$y/x$
2	8	4
5	20	4
7	28	4

Direct variation with $k = 4$.

Inverse: Check if $xy$ is constant.

$x$	$y$	$xy$
2	12	24
3	8	24
6	4	24

Inverse variation with $k = 24$.

Identifying from Graphs

• Direct variation: straight line through the origin.
• Inverse variation: hyperbola (curve approaching the axes).

Tip 1: Find $k$ First

Use a given pair $(x, y)$ to find $k$, then use $k$ to answer the question.

Tip 2: "Proportional" = Direct Variation

"$y$ is proportional to $x$" means $y = kx$.

Tip 3: "Inversely Proportional" = Inverse Variation

"$y$ is inversely proportional to $x$" means $y = k/x$.

Tip 4: Check Both Patterns

If given a table and asked about the relationship, compute both $y/x$ and $xy$ to see which is constant.