How this instrument works
Inverse variation says two quantities are tied together so their product never moves: y = k ⁄ x for a fixed constant k, which is exactly the same statement as x·y = k. Picture a rectangle whose area is pinned at k square units — stretch the width and the height must shrink by precisely the factor needed to keep that area unchanged. That single geometric picture carries the whole idea; every other feature of this relationship follows from the area staying put while the two sides trade off against each other.
Plotted on axes, y = k ⁄ x traces a hyperbola with two separate branches, not the single straight line that direct variation (y = kx) draws through the origin. The two families diverge completely in behavior: direct variation sends its output to zero exactly when the input is zero, while inverse variation is undefined right at zero and instead climbs toward infinity as the input approaches it — the curve bends ever closer to both axes without ever actually touching either one.
The pattern shows up wherever a fixed total gets split between a rate and a duration: a 20-mile route walked at 5 mph takes 4 hours, but jogged at 10 mph takes only 2, because speed and time trade off exactly so their product keeps matching the fixed distance. A mistake worth watching for is treating any downward-sloping line as inverse variation. The line y = -2x + 10 also falls as its input rises, but its product changes at every single point along the way, so it fails the one test that actually defines the relationship.
- Enter the fixed relationship's constant into Constant of variation, k — this is the number that the product of x and y must equal for every pair on the curve.
- Enter your known figure into the x field; it cannot be zero, since dividing by zero has no defined answer.
- Read the result in y = k ⁄ x — that is the value which keeps the product locked at your chosen constant.
- Change the x field and watch the result move the opposite way: multiply the input by any factor and the result divides by that same factor.
- To recover the constant from two known numbers instead, multiply them together and type that product into Constant of variation, k before solving for a third figure.
Worked example — 20 miles at two paces
Suppose a trail covers a fixed distance of 20 miles start to finish. Speed and travel time for that route are an inverse variation with constant of variation k = 20, so y = 20 ⁄ x, where x is the pace in mph and y is the time in hours. Enter k = 20 and x = 5, a brisk walking pace, and the sheet returns y = 4: four hours to cover the 20 miles, because pace times time must always equal 20.
Push the pace to x = 10 mph, a light jog, and the result drops to y = 2 hours — half the time for double the pace, and the product 10 × 2 still equals the fixed 20. That two-for-one trade is the entire content of inverse variation: whatever factor multiplies the input, the result is divided by that same factor, because the underlying product k never moves.
Questions
What does it mean for y to vary inversely as x?
It means y = k ⁄ x for some fixed nonzero constant k, which is the same as saying x·y equals that same number no matter which valid pair of values you pick. As x grows, y shrinks in exactly the proportion needed to keep that product unchanged, and the reverse holds as x shrinks.
How is inverse variation different from direct variation?
Direct variation, y = kx, is a straight line through the origin where both quantities rise and fall together. Inverse variation, y = k ⁄ x, is a curve that never touches either axis, where the two quantities move in opposite directions while their product stays fixed at k.
How do I find the constant of variation from a table of values?
Multiply any matching x and y from the table together: k = x·y. If that product comes out the same for every pair in the table, the data is a genuine inverse variation and k is that shared product; if the product changes from row to row, the relationship is something else.
Why does inverse variation graph as a hyperbola instead of a straight line?
Because solving x·y = k for y traces every point whose input times output equals a fixed value k — geometrically the same as fixing a rectangle's area and letting its width and height trade off. The curve bends away from both axes because neither value can reach zero without breaking that fixed product.
Is a downward-sloping line automatically an inverse variation?
No. Plenty of decreasing relationships, such as y = -2x + 10, are linear rather than inverse; the actual test for inverse variation is that x·y stays exactly constant across every pair of points, not merely that the output falls as the input rises.
What happens to y as x approaches zero or grows very large?
As x approaches zero, y = k ⁄ x grows without bound, which is why zero is excluded from the input. As x grows large instead, y shrinks toward zero but never reaches it — the curve gets arbitrarily close to both axes without ever crossing them.