Skill: Translate figures on a coordinate plane

A translation slides a figure without turning or resizing it. Every point moves the same number of units horizontally and vertically.

The rule

$$(x,y)\rightarrow(x+a,y+b)$$

means:

• Add $a$ to the $x$-coordinate to move left or right.
• Add $b$ to the $y$-coordinate to move up or down.
• A positive number means right or up.
• A negative number means left or down.

Example: Translate triangle $ABC$ with vertices

$$A(1,2),\quad B(4,2),\quad C(2,5)$$

using the rule

$$(x,y)\rightarrow(x-3,y+4).$$

This means move every point $3$ units left and $4$ units up.

Apply the rule to each vertex:

$$\begin{aligned} A(1,2)&\rightarrow(1-3,2+4)=(-2,6),\\ B(4,2)&\rightarrow(4-3,2+4)=(1,6),\\ C(2,5)&\rightarrow(2-3,5+4)=(-1,9). \end{aligned}$$

So the translated triangle has vertices

$$A'(-2,6),\quad B'(1,6),\quad C'(-1,9).$$

To check your work, make sure the same horizontal and vertical changes were used for every point. The triangle keeps the same side lengths, angles, size, and shape; it has only moved.