Stretch the whole plane from the origin by a factor k — every coordinate just multiplies by k.
A photocopier set to 200% keeps a picture's shape but doubles its size; that is scaling. On the coordinate plane, scaling from the origin is the simplest rule of all: pull every point along its ray from O until it sits k times as far away, and its coordinates just multiply — (x, y) ↦ (kx, ky). A k bigger than 1 enlarges, a k between 0 and 1 shrinks, and a negative k throws the image clear across to the opposite side of O. With that, the whole toolkit from this stage is complete: flipping a sign is symmetry, adding a number is translation, multiplying is scaling — three operations, three jobs.
We have already turned points into pairs of numbers (20.1) and watched reflections and slides become arithmetic on those pairs (20.2). This lesson adds the last basic move — the stretch — and shows it is just as clean. Multiplying both coordinates by the same number k is a dilation centered at the origin, the engine behind the similar figures you met in geometry.
Enlarge a photo and something stays the same while something changes. Every angle is untouched — a corner that was square stays square — but every length grows by the same ratio. Two figures related this way are similar: same shape, different size. That common ratio of matching lengths is the scale factor.
In the picture below the big triangle is the small one enlarged by a factor of 2. Side DE is twice AB, side EF is twice BC, side FD is twice CA — one ratio governs them all:
DEAB = EFBC = FDCA = 2.
Similar means same shape, possibly different size. Matching angles stay equal; matching lengths all change by one scale factor. Scaling in coordinates is how we build a similar figure on demand.
Here is the rule, and it could not be simpler. A scaling (also called a dilation) centered at O with factor k sends each point to k times its distance from O along the same ray. In coordinates that means you just multiply both numbers:
(x, y) ↦ (kx, ky).
Why does multiplying do the job? The point (x, y) sits at the end of an arrow from O. Stretching that arrow to k times its length stretches each of its parts — the x-part and the y-part — by the same k, giving (kx, ky). The point moves straight out along its own ray, never sideways.
The origin is the one fixed point: O = (0, 0), and k·0 = 0, so O never moves. It is the anchor every other point is pulled toward or pushed away from. And because every length is multiplied by the same |k| while every angle is preserved, the image is always similar to the original.
Take the triangle with corners (1, 1), (3, 1), (2, 3) and scale from O by k = 2. Multiply both coordinates of each corner by 2:
(1, 1) ↦ (2, 2); (3, 1) ↦ (6, 2); (2, 3) ↦ (4, 6).
Each new corner sits twice as far from O along the very same ray — twice as far across and twice as far up.
Scaling multiplies, it does not add. The origin is the anchor that stays put; every other point slides along its own ray to k times its distance, so (x, y) ↦ (kx, ky).
Drag k across negatives, fractions, and whole numbers. Watch the green image and the slate rays — and read how lengths and area change.
The single number k carries all the information about the stretch. Reading its size tells you whether the figure grows or shrinks; reading its sign tells you which side of O the image lands on.
Size of k. When k > 1 the image is bigger and sits on the same side of O — an enlargement. When 0 < k < 1 the image is smaller, still on the same side — a reduction. When k = 1 nothing moves at all: every point maps to itself.
Sign of k. A negative k does two jobs at once: it scales by |k| and throws the image to the opposite side of O, because each coordinate flips sign as well as changing size. The special case k = −1 scales by |−1| = 1 (no size change) and flips both signs — which is exactly the half-turn about O from 20.2, the point symmetry (x, y) ↦ (−x, −y).
k = ½: (4, 2) ↦ (½·4, ½·2) = (2, 1). Same side of O, half as far out — a shrink.
k = −1: (3, 1) ↦ (−1·3, −1·1) = (−3, −1). Same distance from O, but straight across to the far side — the half-turn.
| scale factor k | what happens | which side of O |
|---|---|---|
| k > 1 | enlarges | same side |
| k = 1 | no change (identity) | same side |
| 0 < k < 1 | shrinks | same side |
| k = −1 | half-turn (point symmetry) | opposite side |
| k < 0 | scales by |k| and flips across O | opposite side |
A negative k is not only a flip. It flips the figure across O and scales it by |k|. So k = −2 makes the image twice as big and on the far side; only k = −1 keeps the size the same.
Every length is multiplied by |k| — that is what "k times as far from O" means for each side. But area is two-dimensional, so it picks up the factor twice: a region scaled by k covers k² times the area. Double the lengths (k = 2) and the area becomes 2² = 4 times as large; halve them (k = ½) and the area becomes (½)² = ¼ as large. The sign of k never affects area, because k² is never negative.
Step back and look at the whole stage at once. Three arithmetic operations act on the address (x, y), and each one does a single geometric job:
The pair (x, y) is the dial each operation turns. Flip the sign and you reflect; add and you slide; multiply and you stretch. Notice the neat pattern: a sign flip is the special multiply by −1, and that is why k = −1 coincides with the half-turn. These three rules are exactly the ones used in the next stage to move whole function graphs around — so it pays to know them cold.
| operation on (x, y) | geometric move | example |
|---|---|---|
| flip a sign | reflection / symmetry | (2, 1) ↦ (−2, 1), (2, −1), (−2, −1) |
| add a constant | translation (slide) | (2, 1) ↦ (2+3, 1+2) = (5, 3) |
| multiply by k | scaling about O | (2, 1) ↦ (2·2, 2·1) = (4, 2) |
Pick an operation on the point P(2, 1) and see the geometric move it names. Three operations, three jobs.
Flip = symmetry, add = translation, multiply = scaling. Every basic move on the plane is one of these three operations acting on the coordinate pair — and a half-turn is just multiplying by −1.
Scaling from the origin is multiplication, plain and clean:
Scale the point (3, 5) from the origin by k = 2.
Multiply both coordinates by k: (2·3, 2·5) = (6, 10).
Scale the point (8, −4) from the origin by k = ½.
(½·8, ½·(−4)) = (4, −2). A negative coordinate still just multiplies — the sign rides along.
Scale (2, 3) by k = −1. Where does it land, and what single move is that?
(−1·2, −1·3) = (−2, −3). Both signs flip with no size change — this is the half-turn about O (point symmetry), the same move as k = −1.
Under a scaling from O the point (5, 0) maps to (15, 0). Find k.
k·5 = 15, so k = 3. (Check the y-part: k·0 = 0 ✓.)
A triangle is scaled from O by k = 3. How do its side lengths and its area change?
Every side length is multiplied by |k| = 3, so lengths × 3. Area is two-dimensional, so it is multiplied by k² = 3² — area × 9.
Which operation on (x, y) produces a translation, and which produces a reflection?
Adding a constant to each coordinate, (x + a, y + b), is a translation (a slide). Flipping a sign, e.g. (x, −y) or (−x, y), is a reflection — symmetry. (Multiplying by k is the third job: scaling.)
Six questions to lock it in. Tap the answer you think is right.
This lesson turns the geometric dilation from Stage 17 into a one-line rule on coordinates: a scaling centered at the origin sends (x, y) to (kx, ky). The payoff is conceptual closure for the whole stage — students see that flipping a sign is symmetry, adding a constant is translation, and multiplying by a factor is scaling. Linking k = −1 back to the half-turn of 20.2 makes the unity concrete: a reflection through O is simply the scale factor −1.
Watch for four classic misconceptions. (1) Adding instead of multiplying — students may compute k + x rather than k·x; insist on "k times as far from O." (2) Forgetting area scales by k², not k — anchor it with the unit-square picture (the figure above shows 4 copies for k = 2). (3) Thinking a negative k only flips — stress it both flips across O and scales by |k|, so only k = −1 keeps the size. (4) Expecting the origin to move — it is the fixed anchor, since k·0 = 0. Have students verify a couple of images by hand and confirm each lands on its ray from O at the right multiple.
8.G.A.3 describe the effect of dilations on two-dimensional figures using coordinates; 8.G.A.4 understand similarity through dilations and rigid motions. G-SRT.A.1 a dilation takes a line to a parallel line (or itself) and multiplies lengths by the scale factor; G-CO.A represent transformations in the plane. Connects backward to dilation of figures and 20.2 symmetry & translation, and forward to transforming function graphs.