Skill: Describe transformations on a coordinate plane

A transformation moves a point or shape on a coordinate plane without changing its size or shape. For example:

• A translation slides a figure.
• A reflection flips a figure across a line, such as an axis.
• A rotation turns a figure around a point, often the origin.

In a transformation, the original figure is called the preimage, and the new figure is called the image. Matching points are called corresponding points.

Example: Triangle $ABC$ has vertices

$$A(1,1), \quad B(4,1), \quad C(1,3).$$

Translate the triangle $2$ units right and $3$ units up. Find the new coordinates.

Step 1: Move each point horizontally.

Moving $2$ units right means add $2$ to each $x$-coordinate:

• $A(1,1)$ becomes $(3,1)$.
• $B(4,1)$ becomes $(6,1)$.
• $C(1,3)$ becomes $(3,3)$.

Step 2: Move each point vertically.

Moving $3$ units up means add $3$ to each $y$-coordinate:

• $(3,1)$ becomes $A'(3,4)$.
• $(6,1)$ becomes $B'(6,4)$.
• $(3,3)$ becomes $C'(3,6)$.

The image of the triangle has vertices

$$A'(3,4), \quad B'(6,4), \quad C'(3,6).$$

The corresponding points are $A$ and $A'$, $B$ and $B'$, and $C$ and $C'$. The triangle has moved, but its side lengths, angles, and overall shape stayed the same. On a coordinate grid, draw the original triangle, move every vertex $2$ units right and $3$ units up, and connect the new vertices.