When two quantities multiply to a fixed amount, one grows as the other shrinks: y = k∕x, with a forbidden spot at x = 0.
A linear function walks at a steady pace — every step of x adds the same amount to y. But suppose instead you have a fixed amount to share: $12 to spend, and the more each item costs the fewer you can buy; a garden bed of fixed area, and the longer you make it the narrower it gets. Now the two quantities don't add in step — they multiply to a constant, so when one doubles, the other halves. That is inverse proportion, written y = k∕x, and this lesson builds it from the ground up: the fixed product xy = k, the definition with k ≠ 0, its three equivalent disguises, how a single pair of numbers pins down k, and the one place x is never allowed to be.
You have already met two ways for variables to pair up. In direct proportion y = kx, and in a linear function y = kx + b, a fixed stride ties x and y together: take one step in x and y always moves by the same fixed amount. Now meet a different kind of tie.
Imagine you have exactly $12 to spend, all of it, on identical items. If each item costs $12 you can buy 1; at $6 each you can buy 2; at $4 each you can buy 3. As the price-each goes up the count goes down — and notice the pattern hiding in plain sight: price × count = 12 every single time.
| count | price each ($) | product |
|---|---|---|
| 1 | 12 | 12 |
| 2 | 6 | 12 |
| 3 | 4 | 12 |
| 4 | 3 | 12 |
| 6 | 2 | 12 |
When the product of two quantities stays fixed — one rises exactly as the other falls — the two quantities are in inverse proportion.
Geometry tells the same story. Take a rectangle of fixed area 12 and let its length L vary. To keep the area at 12 the width must adjust: length 1 → width 12, length 2 → width 6, length 3 → width 4, length 4 → width 3. The taller you make it the narrower it must become, with length × width = 12 at every moment.
Drag the length below. The amber corner of the rectangle traces out a smooth curve — and the green rectangle's area never changes.
A constant product — write it xy = k — is the heart of inverse proportion. Hold onto that equation; everything else flows from it.
We are ready to name it. An inverse-proportion function is
y = kx (k ≠ 0 a constant),
where the constant k is the constant of proportionality.
Its signature behavior follows straight from xy = k: since the product can't change, double x and y is halved; triple x and y becomes a third. For our fixed-area rectangle k = 12, so y = 12∕x. Look at two points on it: at x = 2, y = 6; double x to 4 and y drops to 3 — exactly half, because 4 · 3 is still 12.
The same relationship wears three outfits, and a fluent reader recognizes all of them at a glance:
| form | looks like | read it as |
|---|---|---|
| solved for y | y = k∕x | output is k divided by x |
| a negative power | y = kx⁻¹ | same thing, since x⁻¹ = 1∕x |
| the product form | xy = k | the two multiply to a constant |
Pick whichever outfit is handiest for the job in front of you. Solving for an output? Use y = k∕x. Recognizing the structure in an expression? The power form y = kx⁻¹ helps. Checking a point or finding k? Reach for xy = k.
The product form xy = k is often the quickest test of whether a point lies on the graph: just multiply its coordinates and see if you get k. Is (4, 3) on y = 12∕x? 4 · 3 = 12 ✓ — yes.
Here is the payoff of xy = k: because k = x·y, a single matched pair (x₀, y₀) is enough to pin down the whole function. Multiply the two coordinates to get k, then write y = k∕x. One product fixes one constant, and one constant fixes the entire curve.
Worked example. The graph of an inverse proportion passes through (2, 6). Then
k = x₀·y₀ = 2 · 6 = 12, so y = 12∕x.
Check with another point: does (3, 4) lie on it? 3 · 4 = 12 ✓. It does. Now make your own pair below and watch the curve it forces.
One point fixes the whole inverse proportion, because one product fixes k: k = x₀·y₀, then y = k∕x.
There is one number x is never allowed to be. A denominator can never be zero, so in y = k∕x we must have x ≠ 0. The domain is every real number except 0, written {x | x ≠ 0} or, in interval language, (−∞, 0) ∪ (0, ∞).
Geometrically the curve is born with a slit at the origin: it never crosses the y-axis. And since k ≠ 0, the output y = k∕x is never 0 either, so the range is y ≠ 0 — the curve never touches the x-axis. The two axes are lines the graph approaches but never reaches.
x = 0 has no output at all — that single missing value is exactly why the graph comes in two pieces rather than one, the discovery waiting in the next lesson.
Inverse proportion is the partner of the steady linear stride: instead of a fixed sum-of-steps, two quantities keep a fixed product.
Is y = 8∕x an inverse-proportion function? If so, what is k?
Yes — it has the form y = k∕x with a nonzero constant, so k = 8.
The graph of an inverse proportion passes through (3, 5). Find k and the formula.
k = x₀·y₀ = 3 · 5 = 15, so y = 15∕x.
For y = 12∕x, find y when x = 4 and when x = −6.
At x = 4: y = 12∕4 = 3. At x = −6: y = 12∕(−6) = −2. (Check the products: 4·3 = 12, (−6)(−2) = 12 ✓.)
Rewrite xy = 20 in the form y = k∕x. What is k?
Divide both sides by x (x ≠ 0): y = 20∕x, so k = 20.
For y = 6∕x, if x doubles from 2 to 4, what happens to y?
y goes from 6∕2 = 3 to 6∕4 = 1.5 — it halves (3 → 1.5), because the product xy = 6 must stay fixed.
What is the domain of y = 7∕x, and why?
x ≠ 0 — you can't divide by zero, so x = 0 has no output. In interval form, (−∞, 0) ∪ (0, ∞).
Six questions to lock it in. Tap the answer you think is right.
The big shift here is from additive to multiplicative pairing. A linear function keeps a constant difference per step; an inverse-proportion function keeps a constant product. Anchoring every claim to the one equation xy = k — read three ways — lets a learner derive the rest rather than memorize it: double x must halve y; one point gives k = x₀·y₀; x can never be 0 because a denominator can't be 0.
The misconception to watch is thinking inverse proportion adds like a line when it actually multiplies — and a learner who writes y = kx (direct) for y = k∕x (inverse) has swapped the two families. Two guardrails fix this: insist on the product test (do x and y multiply to a constant, or differ by a constant?), and never let k = 0 or x = 0 slip past — both break the definition. The forbidden spot x = 0 is not a technicality; it is the reason the graph splits in two.
Common Core. This lesson supports 7.RP.A.2 (recognize and represent proportional relationships — here the inverse case), 8.F.A.1 / F-IF.A.1 (a function assigns exactly one output to each input; the domain excludes x = 0), and A-CED.A.2 (write y = k∕x from a described relationship).