Inverse Variation Calculator
Work with inverse variation relationships of the form y = k/xⁿ — find the constant of variation, solve for an unknown value from a known pair, check whether a data table actually represents inverse variation (versus direct variation or no pattern at all), or graph the curve and evaluate it at a chosen x. Every mode shows the full substitution, a diagram of what's actually happening, and a callout explaining what the result means.
Background
In an inverse variation, one quantity increases exactly as fast as the other decreases, following the rule y = k/xn for a fixed constant k (the constant of variation) and a whole-number power n — usually n=1, but n=2 ("inverse square") and n=3 ("inverse cube") show up constantly in physics and geometry. Unlike direct variation, where doubling x doubles y, doubling x in an inverse variation shrinks y instead.
How to use this calculator
- Choose Find the Constant to work out k from a single known (x, y) pair, for a chosen power n.
- Choose Solve for a Value to use a known pair to find a new y (given a new x) or a new x (given a new y) — the classic "varies inversely" word problem.
- Choose Identify the Pattern to feed in a small table of (x, y) pairs and see whether it matches direct variation, inverse variation of some power, or neither.
- Choose Graph & Explore to see the curve for a given k and n, its asymptotes and range, and optionally its value at a chosen x.
- Click Calculate to see the diagram, the full step-by-step math, and a callout explaining what the result actually means.
How Inverse Variation Works
An inverse variation is a relationship of the form y = k/xn, where k is a fixed constant (the constant of variation) and n is a whole-number power — most often 1, but 2 and 3 show up in real applications too.
The constant of variation is found by rearranging: k = xn·y. Plug any valid (x, y) pair from the relationship in, and you'll get the same k every time.
That's the defining "inverse" behavior: as x increases, y decreases (and vice versa) — the opposite of direct variation, where they move together.
Because every valid pair shares the same constant, two points on the same relationship satisfy x1n·y1 = x2n·y2 = k. That's how an unknown value gets solved without any extra formulas.
The domain always excludes x = 0, since k/0n is undefined — this shows up as a vertical asymptote at x = 0 on every inverse variation graph.
As |x| grows without bound, y shrinks toward — but never reaches — 0. That's a horizontal asymptote at y = 0, true for every inverse variation regardless of k or n.
When n is even (inverse square, inverse fourth, etc.), xn is always positive, so y always carries the same sign as k. When n is odd, y's sign flips along with x's sign.
To tell inverse variation apart from direct variation (y = kx) or no clean pattern at all, check which quantity stays constant across a data table: y/x constant means direct; xn·y constant for some n means inverse of that order; nothing constant means neither.
Formulas & Equations Used
General inverse variation: y = k / xⁿ
Constant of variation: k = xⁿ · y
Relating two points on the same curve: x1ⁿ · y1 = x2ⁿ · y2
Solving for a new x: x2 = (k / y2)^(1/n) — real only when k/y2 ≥ 0 for even n
Named cases: n=1 → inverse variation, n=2 → inverse square variation, n=3 → inverse cube variation
Example Problems & Step-by-Step Solutions
Example 1 — Finding the constant (basic inverse)
y varies inversely as x. When x=6, y=8. Find k.
Step: k = x·y = 6 × 8 = 48.
Result: k = 48, so y = 48/x.
Example 2 — Finding the constant (inverse square)
y varies inversely as the square of x. When x=3, y=5. Find k.
Step: k = x²·y = 9 × 5 = 45.
Result: k = 45, so y = 45/x².
Example 3 — Solving for a new value
y varies inversely as x. When x=4, y=15. Find y when x=6.
Step: k = 4 × 15 = 60. y = 60/6.
Result: y = 10.
Example 4 — Solving for a new x (even power)
y varies inversely as the square of x. When x=2, y=20. Find x when y=5.
Step: k = 4 × 20 = 80. x² = 80/5 = 16.
Result: x = ±4 — both are valid since squaring either gives 16.
Example 5 — Identifying inverse variation from a table
Data: (2,10) (4,5) (5,4) (10,2). Is this direct, inverse, or neither?
Step: Check x·y: 20, 20, 20, 20 — constant.
Result: Inverse variation, k = 20.
Example 6 — Spotting inverse square in a table
Data: (1,90) (2,22.5) (3,10). Is this ordinary inverse variation?
Step: x·y gives 90, 45, 30 — not constant. x²·y gives 90, 90, 90 — constant.
Result: Inverse square variation, k = 90, not plain inverse variation.
Example 7 — Graphing and asymptotes (odd power)
For y = 8/x, describe the asymptotes and range.
Step: Vertical asymptote x=0, horizontal asymptote y=0. n=1 is odd, so y's sign follows x's sign.
Result: Range is all y ≠ 0, split across two branches in opposite quadrants.
Example 8 — Range restriction with an even power
For y = −12/x², describe the range.
Step: n=2 is even, so x² ≥ 0 always; k = −12 is negative.
Result: Range is y < 0 only — the curve lives entirely below the x-axis, no matter the sign of x.
Frequently Asked Questions
What's the difference between direct and inverse variation?
Direct variation is y = kx: both quantities increase or decrease together. Inverse variation is y = k/xⁿ: one increases exactly as the other decreases.
Why is x=0 always excluded from the domain?
Dividing by 0 (or 0 raised to any power) is undefined, so every inverse variation has a vertical asymptote at x=0 regardless of k or n.
What is an "inverse square law" and where does it show up?
It's the n=2 case, y = k/x². It describes phenomena that spread out over a growing sphere's surface as distance increases, such as light intensity, gravitational force, and signal strength.
Why does an even power like n=2 restrict y to one sign?
Because x² (or any even power) is always ≥ 0 for a real x, so k/xⁿ always carries the same sign as k, no matter which side of zero x sits on.
How can I tell if a table of data represents inverse variation?
Check whether x·y (or xⁿ·y for a higher power) stays the same for every pair. If it does, that value is the constant of variation and confirms an inverse relationship of that order.
Can the constant of variation k be negative?
Yes. A negative k just flips which side of the x-axis (or which quadrants) the curve occupies — the asymptotes and the inverse relationship still work the same way.
What happens if solving for x asks for an even root of a negative number?
There's no real solution. That means the given y-value can't actually occur on that curve, since xⁿ can never carry a sign different from k's when n is even.
Does this calculator handle variation with more than one variable (joint variation)?
No — this tool covers the standard single-variable case, y = k/xⁿ. Joint variation, where y depends on the product of several variables at once, needs its own dedicated calculator.
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