How this instrument works
Direct variation is the plain statement that y is always some fixed multiple of x: y = kx for one constant k that never changes across every pair of values in the relationship. Divide any y by its matching x and the same number falls out every time — that constant is what these fields solve for. Graphed, a direct variation is a straight line through the origin, because setting x to zero forces y to zero as well; there is no starting offset to add in.
The defining feature is the ratio, not just the rate of change. A general linear relationship y = mx + b shares the same constant slope as direct variation, but the added b shifts the whole line off the origin, so y ⁄ x stops being constant except in the single case b = 0. Direct variation is exactly that special case: pick any two pairs that satisfy it and y₁ ⁄ x₁ equals y₂ ⁄ x₂ exactly, which is the cross-multiplication logic behind 'if 2 apples cost $6, what do 5 apples cost.'
Nothing requires k to be positive: y = −2x is direct variation just as much as y = 2x, since the ratio y ⁄ x still holds rock-steady at −2, it only means y falls as x rises. The one genuine edge case is x₁ = 0, where the ratio is undefined — division by zero — and this sheet checks for that before it will report a constant. Physics leans on the same shape often: force on an ideal spring, distance covered at constant speed, and Ohm's law, V = IR, are all direct variation dressed in different letters.
- Enter your known pair into Known x₁ and Known y₁ — any matched values that already satisfy y = kx will do.
- Enter the point you want to project to in New x₂.
- Read Constant of variation, k — the fixed ratio y₁ ⁄ x₁ shared by every pair in the relationship.
- Read y₂ = k·x₂ for the y-value that keeps that same ratio at the new x you entered.
- If Known x₁ is left at zero, the sheet flags it — the ratio y₁ ⁄ x₁ has nothing to divide into.
Worked example — a spring stretched from (2, 6) to x = 5
A spring is known to obey direct variation: hang a 2 kg weight and it stretches exactly 6 cm, with no initial slack to subtract first. That is Known x₁ = 2 and Known y₁ = 6, so the constant of variation is k = y₁ ⁄ x₁ = 6 ⁄ 2 = 3 — every kilogram of load adds 3 cm of stretch, and the relationship holds all the way back to (0, 0).
Hang a heavier 5 kg weight instead — New x₂ = 5 — and the same fixed ratio applies: y₂ = k·x₂ = 3 × 5 = 15 cm. The 2 kg reading was not special; feed in x₁ = 1 and y₁ = 3 and k still returns 3, because direct variation pins down one ratio for the whole line, not just one measured pair.
Questions
What does it mean for y to vary directly with x?
It means y is always exactly k times x for one fixed constant k — double x and y doubles, halve x and y halves, and the pair (0, 0) always satisfies the relation. Graphed, direct variation is a straight line through the origin with slope k; a line crossing the y-axis anywhere else, however small the offset, is only linear, not directly variational.
How is direct variation different from a linear function with a y-intercept?
A linear function y = mx + b only reduces to direct variation when b = 0. With any nonzero intercept, doubling x no longer doubles y — y = 2x + 1 gives y = 3 at x = 1 but y = 5 at x = 2, not 6. Direct variation is the special case where the line is pinned through the origin, so the ratio y ⁄ x stays fixed for every point, not merely the slope between points.
Can the constant of variation k be negative?
Yes — k only needs to be a fixed, nonzero constant, and a negative k still describes direct variation, just with y decreasing as x increases and vice versa. With k = −2, meaning y = −2x, doubling x still exactly doubles the size of y; it only flips the sign. What direct variation rules out is a changing ratio, not a particular sign for k.
Why can't Known x₁ be zero?
Because the constant of variation is computed as k = y₁ ⁄ x₁, and dividing by zero has no defined result. If x₁ really were zero, direct variation already forces y₁ to be zero too, since every such line passes through the origin — so that pair carries no information about the ratio, and a nonzero x is needed to read off k.
How is direct variation related to inverse variation?
They are opposite behaviors built from the same idea of a fixed constant. Direct variation is y = kx, where y grows as x grows; inverse variation is y = k ⁄ x, where y shrinks as x grows so that the product xy, not the ratio y ⁄ x, stays constant. The common mixup is checking the wrong quantity — see whether a ratio or a product is what holds steady.
What are some real examples of direct variation?
Distance at a constant speed (distance = speed × time), the cost of identical items (total = price × quantity), and Hooke's law for an ideal spring within its elastic limit (force = stiffness × extension) are all direct variation, because doubling one quantity doubles the other with no fixed offset involved.