Direct and Inverse Variation

Direct variation: $y = kx$

$y$ and $x$ change in the same direction. Double $x$ → double $y$. The graph is a straight line through the origin. $k$ is the constant of proportionality.

Example: distance $= \text{speed} \times \text{time}$. At constant speed, doubling time doubles distance.

Inverse variation: $y = k/x$

$y$ and $x$ change in opposite directions. Double $x$ → halve $y$. The graph is a hyperbola. The product $xy = k$ is always constant.

Example: If 6 workers finish a job in 4 hours, then 12 workers finish in 2 hours. More workers → less time.

How to tell them apart

• Direct: $y/x = k$ (constant ratio)
• Inverse: $xy = k$ (constant product)

Try it in the visualization

Both functions are graphed side by side. Adjust $k$ and trace how $y$ responds to changes in $x$. For direct variation, the line steepens; for inverse, the hyperbola shifts. $y= k/x$. $y$ and $x$ change in opposite directions. The constant $k$ is the constant of proportionality.