Transformations

Four ways to move a shape:

• Translation: slide it. Every point moves the same distance in the same direction.
• Reflection: flip it over a line, like a mirror.
• Rotation: turn it around a point by some angle.
• Dilation: resize it from a centre point by a scale factor.

The first three are rigid: the shape keeps its size and angles, so the image is congruent to the original. Only dilation changes size, producing a similar but not congruent figure.

Coordinate rules worth knowing: reflecting over the x-axis takes (x, y) to (x, −y); over the y-axis it becomes (−x, y). A dilation by k from the origin gives (kx, ky).

Translations, reflections and rotations are isometries: they preserve distance, and therefore angle measure, area and congruence. Dilations preserve angle and shape but scale distances by the factor k, producing similarity rather than congruence.

In coordinates: translation is (x, y) → (x + h, y + k); reflection over y = x is (x, y) → (y, x); rotation by 90° anticlockwise about the origin is (x, y) → (−y, x); dilation centred at the origin is (x, y) → (kx, ky).

These same maps reappear as function transformations in Algebra 2, where f(x − h) + k is a translation, −f(x) and f(−x) are reflections across the axes, and a·f(x) is a vertical dilation. The counterintuitive part there — that f(x − h) shifts *right* by h — is easier to accept once you notice that the substitution changes the input needed to reach a given output, not the output itself.