Two quantities locked in step: y = kx, a straight line through the origin, with k the fixed stride.
Buy apples at a fixed price and the cost rides along with the weight — double the pounds, double the price. That "locked-together" change is the plainest function there is, and on the coordinate plane it draws the cleanest picture of all: a straight line that passes through the origin. In this lesson we meet the direct-proportion function y = kx, read its k as a fixed stride — how much y climbs for every 1 that x gains — and watch how the sign and size of that one number set the whole line's direction and steepness. Everything here is built from the coordinate plane and the idea of a function you met in Stage 20; we are simply meeting the first concrete family of functions, the one every other linear rule is built on.
Recall the heart of a function from Stage 20: when fixing one quantity settles another, we say y is a function of x, written y = f(x). You choose x — the input — and the rule hands back exactly one output y. On grid paper this feels physical: walk a point left and right along a line, and as its x changes, its height y is dragged along with it. Set x, and y is determined.
The figure below shows one such point on a line. Read it as a machine: dial in an x at the bottom, follow the dashed guide up to the line, then across to the y-axis to read the output. No matter where you stop, there is exactly one height — that is what makes it a function.
A rule is a function when each input x produces exactly one output y. We write y = f(x) and read it "y is a function of x."
Now meet the simplest function of all. Apples sell at $3 a pound. Then 1 lb costs $3, 2 lb costs $6, 3 lb costs $9 — double the weight and you double the price. Notice what stays fixed: the ratio price ÷ weight is always the same number,
31 = 62 = 93 = 3.
Because that ratio never budges, the price p is always 3 times the weight w. Write it as a clean rule: p = 3w. Whenever one quantity is a fixed multiple of another — y is always (a constant) times x — we say the two are in direct proportion. Slide the running point along the line in the band below and watch the ratio hold steady at 3.
Two quantities are in direct proportion when one is a constant multiple of the other: y = kx. The constant k is the unchanging ratio y ÷ x.
Here is the definition we will lean on for the rest of the stage. A direct-proportion function is a rule of the form
y = kx (k ≠ 0, a constant),
and the number k is called the constant of proportionality. The cleanest way to feel what k does is to read it as a stride: every time x goes up by 1, y changes by exactly k. Check it with algebra — step from x to x + 1:
at x: y = kx; at x + 1: y = k(x + 1) = kx + k. The change is (kx + k) − kx = k.
So for apples, p = 3w with k = 3 means simply "+$3 for each extra pound." The slope triangle below makes the stride visible: a run of 1 across always comes with a rise of k up to the line.
In y = kx, the constant k is the stride: each unit increase in x raises y by exactly k. We require k ≠ 0 (otherwise y is stuck at 0 and there is no proportion).
Two points fix a line, and y = kx hands us one of them for free. Put x = 0: y = k·0 = 0, so the line always passes through the origin O(0, 0). That is the signature of a direct proportion — its graph goes straight through O. We need only one more point, and the easiest choice is x = 1, which lands at (1, k).
Take y = 2x as a worked example. Plot O(0, 0) and (1, 2), lay a ruler across the two, and extend in both directions — that straight line through the origin is the graph. The slider below lets you set k and watch the line pivot about O.
You only need one point besides the origin. Pick one with whole-number coordinates — for y = kx the point (1, k) is always handy — so the ruler lands exactly on the grid.
That one number k tells you everything about how the line sits. Look at its sign first:
k > 0: as x grows, y grows too — the line climbs to the upper right, travelling through Quadrants Ⅰ and Ⅲ. k < 0: as x grows, y falls — the line slides down to the lower right, through Quadrants Ⅱ and Ⅳ.
Now its size: the larger |k| is, the more y moves for each step across, so the steeper the line. Compare three lines through the origin — y = 2x (steep, rising), y = ½x (gentle, rising), and y = −2x (steep, falling):
Use the same slider once more, now sweeping k through positive and negative values, to feel the verdict change.
| sign / size of k | direction | quadrants | steepness |
|---|---|---|---|
| k > 0 | rises ↗ | Ⅰ & Ⅲ | — |
| k < 0 | falls ↘ | Ⅱ & Ⅳ | — |
| larger |k| | — | — | steeper |
k is the line's slope: its sign decides rise or fall (k > 0 rises through Ⅰ & Ⅲ, k < 0 falls through Ⅱ & Ⅳ), and its size |k| decides steepness — bigger |k|, steeper line.
Two quantities are in direct proportion when one is a constant multiple of the other: y = kx with k ≠ 0. Its graph is a straight line through the origin — set x = 0 and y = 0, so O(0, 0) is always on it. The constant k, the constant of proportionality, is the line's slope and its stride: each step of 1 in x changes y by exactly k. The sign of k sets direction — k > 0 rises (through Ⅰ & Ⅲ), k < 0 falls (through Ⅱ & Ⅳ) — and the size |k| sets steepness, larger meaning steeper. To draw one, plot O and one more lattice point such as (1, k) and rule the line. Next lesson (21.2) gives that line a starting point b and lifts it off the origin into the general linear function y = kx + b.
Is y = 5x a direct proportion? If so, what is the constant of proportionality k?
Yes — it has the form y = kx (a constant times x), so it is a direct proportion with k = 5.
A direct proportion y = kx passes through the point (2, 10). Find k.
Substitute the point: 10 = k·2, so k = 10 ÷ 2 = 5. The function is y = 5x.
For y = −3x, find y when x = 4, and say whether the line rises or falls.
y = −3·4 = −12. Since k = −3 < 0, the line falls (through Quadrants Ⅱ and Ⅳ).
The graph of any direct proportion always passes through which special point? Why?
The origin (0, 0): putting x = 0 into y = kx gives y = k·0 = 0, so (0, 0) is always on the line.
Which line is steeper, y = 4x or y = ½x?
Compare the sizes of the slopes: |4| = 4 is larger than |½| = ½, so y = 4x is steeper.
8 pens cost $12. Write the cost c as a function of the number n of pens, then find the cost of 5 pens.
Cost is in direct proportion to count: the price per pen is k = 12 ÷ 8 = 1.5, so c = 1.5n. For 5 pens, c = 1.5·5 = $7.50.
Six questions to lock it in. Tap the answer you think is right.
This lesson opens the linear-functions strand with its plainest member, the direct-proportion function y = kx. The big idea is that a constant ratio between two quantities shows up on the coordinate plane as a single, unmistakable shape: a straight line through the origin whose steepness is the constant of proportionality. Connecting the unit rate ("$3 per pound," "1.5 dollars per pen") to the slope of that line is the key skill — it is the same number wearing two hats. Encourage learners to read k aloud as "for every 1 more x, y changes by k."
Watch for three common misconceptions. First, students often think any straight line is a direct proportion — stress that it must pass through the origin (y = 2x + 1 is a line but not a proportion). Second, they confuse the point (1, k) with the slope value k; the point is where you read the slope, not the slope itself. Third, they forget the requirement k ≠ 0 — with k = 0 the "line" collapses onto the x-axis and there is no proportion to speak of.
This material aligns with US Common Core standards 7.RP.A.2 (recognize and represent proportional relationships; identify the constant of proportionality), 8.EE.B.5 (graph proportional relationships and interpret the unit rate as the slope), and 8.F.A.3 / 8.F.B.4 (y = kx is a linear function; determine and interpret its rate of change).