Pick up a paper triangle and move it across the table. Slide it, flip it over, spin it around — through all of that, it is still the same triangle. Same side lengths, same angles, same shape. In geometry we give those movements a name: transformations. And once you place the shape on a coordinate grid, each movement becomes a tidy little rule that tells you exactly where every point lands.
That is the whole magic of this topic. You do not have to redraw a shape point by point and hope it looks right. You learn three rules, and the grid does the bookkeeping for you.
What a transformation actually is
A transformation is a rule that takes every point of a shape and sends it to a new place. The shape you start with is the pre-image; the shape you end up with is the image.
The key word is every. A transformation is not a vague "move it over there" — it is a precise instruction that applies to each and every point. We write a point as \((x, y)\), and the rule tells us what new point it becomes. For example, the rule \((x, y) \rightarrow (x + 4,\ y - 1)\) reads as: "take any point, push it 4 to the right and 1 down."
Three of these movements keep the shape's size and form completely intact. They are the heart of this lesson: translation (a slide), reflection (a flip), and rotation (a turn). Try all three on the same shape below, and watch the rule update as you do.
Press each button and read the \((x, y)\) rule it prints — that rule is the whole idea in one line.
Translation — the slide
A translation slides the shape without turning or flipping it. Every point moves the same distance in the same direction, like sliding a book straight across a desk. Nothing tilts; nothing reverses.
Because the motion is identical for every point, the rule is simply "add the same amounts to the coordinates":
\[ (x,\ y) \rightarrow (x + a,\ y + b) \]Here \(a\) is the horizontal shift (right if positive, left if negative) and \(b\) is the vertical shift (up if positive, down if negative). That is it — translation is just addition.
- The rule says add \(4\) to \(x\) and subtract \(1\) from \(y\): \((x, y) \rightarrow (x + 4,\ y - 1)\).
- New \(x\)-coordinate: \(2 + 4 = 6\).
- New \(y\)-coordinate: \(3 - 1 = 2\).
The point lands at \((6, 2)\) — four steps right and one step down from where it started.
Reflection — the flip
A reflection flips the shape across a line, called the line of reflection, as if that line were a mirror. The image is the same distance from the mirror as the original, just on the other side. Reflections reverse orientation — your right hand becomes a left hand in the mirror.
Which coordinate changes depends on the mirror line:
- Over the \(x\)-axis: \((x,\ y) \rightarrow (x,\ -y)\). The \(x\) stays put; the height flips sign.
- Over the \(y\)-axis: \((x,\ y) \rightarrow (-x,\ y)\). The height stays; left and right swap.
- Over the line \(y = x\): \((x,\ y) \rightarrow (y,\ x)\). The two coordinates simply trade places.
In words To reflect across an axis, ask which axis is the mirror, then flip the sign of the other coordinate. Mirror is the \(x\)-axis → flip \(y\). Mirror is the \(y\)-axis → flip \(x\). The coordinate that lives on the mirror line never moves.
- The mirror is the \(x\)-axis, so the rule is \((x, y) \rightarrow (x,\ -y)\).
- The \(x\)-coordinate is unchanged: it stays \(2\).
- The \(y\)-coordinate flips sign: \(3\) becomes \(-3\).
The image is \((2, -3)\) — the same horizontal position, but now below the axis instead of above it, an equal distance away.
Rotation — the turn
A rotation turns the shape about a fixed point. In this lesson that point is always the origin, \((0, 0)\). Think of pinning the grid at the origin and spinning the whole shape around the pin. We measure the turn in degrees, and "counterclockwise" (the same way angles grow) is the standard positive direction.
Two turns come up again and again:
- 90° counterclockwise: \((x,\ y) \rightarrow (-y,\ x)\). Swap the coordinates, then negate the new first one.
- 180°: \((x,\ y) \rightarrow (-x,\ -y)\). Both signs flip — the point lands straight across the origin.
The 180° rule is easy to feel: half a full turn drops every point to the exact opposite side of the origin. The 90° rule looks stranger, so let's see it in action.
- The rule for a 90° counterclockwise turn is \((x, y) \rightarrow (-y,\ x)\).
- Here \(x = 4\) and \(y = 1\).
- New first coordinate: \(-y = -1\). New second coordinate: \(x = 4\).
The image is \((-1, 4)\). The point has swung a quarter-turn from the lower right up toward the top — exactly what a counterclockwise spin should do.
Tip — turn your paper. If a rotation rule ever slips your mind, sketch the point, then physically rotate the page a quarter or half turn and read off where it ended up. The rule should match what your eyes see. For a 90° counterclockwise turn, a point on the right edge always swings up to the top.
Why these three keep the shape congruent
Translation, reflection and rotation share a special quality: they never stretch, shrink or distort. Distances between points stay the same, and angles stay the same. We call any such motion a rigid motion (or isometry) — "rigid" because the shape behaves like a solid object you are simply repositioning.
Because lengths and angles are preserved, the image is an exact copy of the pre-image. We say the two figures are congruent: same size, same shape, possibly just sitting in a new place or facing a new direction.
One that breaks the rule: dilation
Not every transformation is rigid. A dilation scales a shape larger or smaller from a center point — like a projector zooming an image. Multiply both coordinates by a scale factor \(k\): the rule is \((x, y) \rightarrow (kx,\ ky)\) for a dilation centered at the origin.
A dilation keeps all the angles the same, so the shape still looks like itself — but the side lengths change. The result is similar, not congruent. Use \(k = 2\) and every length doubles; use \(k = \tfrac{1}{2}\) and every length halves. That is the line that separates our three rigid motions from a dilation: rigid motions preserve size, a dilation changes it.
Practice
Try each one yourself, then reveal the full solution.
1. Translate the point \((1, 5)\) by the rule \((x + 3,\ y - 2)\). Where does it land?
Apply the rule \((x, y) \rightarrow (x + 3,\ y - 2)\) to \((1, 5)\):
New \(x = 1 + 3 = 4\); new \(y = 5 - 2 = 3\).
The image is \((4, 3)\) — three steps right and two steps down.
2. Reflect the point \((4, -2)\) over the \(x\)-axis.
The mirror is the \(x\)-axis, so the rule is \((x, y) \rightarrow (x,\ -y)\): keep \(x\), flip the sign of \(y\).
The \(x\)-coordinate stays \(4\). The \(y\)-coordinate \(-2\) flips to \(-(-2) = 2\).
The image is \((4, 2)\) — the same horizontal position, reflected to the other side of the axis.
3. Rotate the point \((3, 1)\) by 90° counterclockwise about the origin.
The 90° counterclockwise rule is \((x, y) \rightarrow (-y,\ x)\): swap the coordinates, then negate the new first one.
With \(x = 3\) and \(y = 1\): new first coordinate \(= -y = -1\); new second coordinate \(= x = 3\).
The image is \((-1, 3)\).