What's the Difference Between Direct and Inverse Variation?

Direct variation means two variables change in the same proportional ratio, while inverse variation means their product remains constant. In the standard cases, direct variation has the form $y=kx$, whereas inverse variation has the form $y=\frac{k}{x}$.

Here, $k$ is the constant of variation, and $k\neq 0$.

To identify the model from a table, test the relevant quantity for every data pair. If (y/x) is constant, the relationship is direct; if (xy) is constant, the relationship is inverse. For example, the pairs $(2,10)$, $(4,5)$, and $(8,2.5)$ are inverse because each product equals $20$. The ratios are not constant.

The graph also provides evidence. A direct-variation graph is a straight line through the origin; an inverse-variation graph approaches both axes as asymptotes and is undefined at $x=0$. The sign of $k$ determines the relevant quadrants. However, a graph alone should not replace the algebraic constant test. In context, restrict the domain to meaningful values, such as positive time or distance.

For direct variation, suppose $y$ varies directly with $x$ and $y=12$ when $x=3$. Then

$$k=\frac{y}{x}=\frac{12}{3}=4,$$

so the model is $y=4x$. When $x=6$, $y=4(6)=24$.

For inverse variation, suppose $y$ varies inversely with $x$ and $y=12$ when $x=3$. Then

$$k=xy=3(12)=36,$$

so the model is $y=\frac{36}{x}$. When $x=9$, $y=\frac{36}{9}=4$.

A common misconception is that inverse variation simply means that one variable decreases as the other increases. That is not sufficient: inverse variation specifically requires (xy) to remain constant. A decreasing linear relationship, such as $y=20-2x$, is not inverse variation.

This is part of SL 2.5, Modelling functions in IB Mathematics: Applications and Interpretation. In an exam, identify the relationship, calculate $k$, write the complete model, and then substitute the requested value. Use your GDC to check values, but show the equation and setup to secure method marks.