Transformations on the Coordinate Plane

People have been moving shapes around for thousands of years—think of tile patterns on ancient temple floors, or the repeating designs on a quilt. But it wasn't until mathematicians created the coordinate plane (that grid with an x-axis and y-axis) that we could describe those moves with numbers. Here's a quick look at the key moments.

So the big question this lesson answers is: how do we use coordinate rules to describe exactly what happens to a shape when we slide it, flip it, spin it, or resize it?

A transformation is a rule that takes every point of a shape and moves it to a new location. In this lesson we study four types. The first three—translations, reflections, and rotations—produce a shape that is the exact same size and shape as the original. We call those rigid motions (or isometries). The fourth, dilation, changes the size but keeps the shape. Let's meet them.

The diagram below shows one triangle going through all four transformations. Study each colored shape and pay attention to how the coordinates change. The original triangle (in cyan) has vertices at A(1, 1), B(4, 1), and C(2, 4).

In the diagram, the cyan triangle is the original. The violet dashed triangle shows a translation—every vertex moved 5 units left and 3 units down. The pink dashed triangle shows a reflection over the y-axis—the x-coordinates flipped sign while the y-coordinates stayed the same. Notice that both new triangles are the same size and shape as the original. That's because translations and reflections are rigid motions.

Here's the exciting part: each transformation has a coordinate rule—a formula you can apply to any point (x, y) to find where it lands. Let's go through each one.

For example, a translation of (−3, 5) means slide every point 3 units left and 5 units up. If a vertex is at (2, 1), it moves to (2 + (−3), 1 + 5) = (−1, 6). Every point shifts by the exact same amounts, so the shape doesn't change at all.

When you reflect over the x-axis, the y-coordinate flips sign because the point jumps to the other side of the horizontal line. Over the y-axis, the x-coordinate flips. Over the line y = x, the two coordinates swap places.

A 90° counterclockwise rotation about the origin sends (3, 2) to (−2, 3). You negate the old y-value and swap positions. A 180° rotation negates both coordinates—it's like doing two 90° turns in a row.

With a scale factor of k = 2, the point (3, 1) becomes (6, 2). Every distance from the origin doubles. The shape looks the same but is twice as big. A factor of k = ½ would cut each coordinate in half, shrinking the shape to half its size.

Rotations and dilations can be tricky because they change coordinates in less obvious ways than sliding or flipping. The diagram below focuses on a 90° counterclockwise rotation and a dilation with scale factor 2, both centered at the origin.

Look at the amber (rotated) square. Each vertex followed the rule (x, y) → (−y, x). For instance, the original vertex (3, 1) became (−1, 3)—you negate the y-value to get the new x, and the old x becomes the new y. The whole square spun 90° counterclockwise but stayed the same size.

Now look at the green (dilated) square. With a scale factor of 2, every coordinate doubled. The side length went from 2 units to 4 units. The shape is still a square, but it's bigger and farther from the origin. That's why dilation is not a rigid motion—it changes the size.

Let's walk through a complete problem step by step.

Which transformations keep size the same? Which keep orientation? This table gives you a side-by-side look.

Notice that reflection is the only rigid motion that reverses the orientation. That means if the original triangle's vertices went in a clockwise order, the reflected triangle's vertices go in a counterclockwise order. A good way to remember this: when you look at your reflection in a mirror, the text on your shirt appears backwards—that's reversed orientation!

The transformation rules you've learned here are the foundation for some powerful ideas in higher math. In high school geometry, you'll use sequences of transformations to prove that two shapes are congruent or similar. Instead of measuring every side and angle, you can show there is a series of translations, rotations, reflections, and/or dilations that maps one shape perfectly onto the other.

In algebra, you'll see transformations applied to graphs of functions—shifting a parabola up (translation), flipping it upside down (reflection), or stretching it (dilation). The same vocabulary and logic carry over directly.