Direct variation refers to a relationship between two variables where when one variable increases, the other increases by the same factor. The relationship between the two variables can be modeled by the equation y = kx, where k is a constant of proportionality. The ratio of y to x will always equal k.
y = 4x
Hannah is growing a plant. When Hannah first planted the plant's seed, the plant had a height of 0. For two variables to vary directly, their graph must pass through the origin. The plant's height (y) increases by 4 inches for every week (x) that has elapsed, so its height varies directly with its age. In the table above, x is time in weeks, and y is the height of the plant.
When someone says that y varies directly with x, or that y is directly proportional to x, they mean that y and x exhibit direct variation.
See also constant of proportionality.