On the four-quadrant coordinate plane you can move whole shapes, not just points. The simplest move is a slide, called a translation.
Why it works
A translation slides every point the same distance in the same direction. Sliding a paper triangle across a desk doesn’t stretch it, turn it or flip it. So side lengths, angle measures and parallel lines all stay the same; only the position changes.
Key idea
Translating $a$ units right (left if $a$ is negative) and $b$ units up (down if $b$ is negative) follows the rule $$(x, y) \to (x + a, y + b)$$
Worked examples
Worked example: Move a point
Translate $A(2, -1)$ 5 units left and 3 units up.
- Left 5 changes $x$: $2 - 5 = -3$.
- Up 3 changes $y$: $-1 + 3 = 2$.
Answer: $A'(-3, 2)$
Worked example: Find the rule
A translation maps $P(1, 4)$ to $P'(6, 2)$. Write its rule.
- Change in $x$: $6 - 1 = 5$. Change in $y$: $2 - 4 = -2$.
Answer: $(x, y) \to (x + 5, y - 2)$: 5 right and 2 down
Worked example: Move a whole figure
Translate the triangle with vertices $(0, 0)$, $(3, 0)$, $(0, 2)$ by $(x, y) \to (x - 2, y + 4)$.
- Apply the rule to each vertex: $(0, 0) \to (-2, 4)$, $(3, 0) \to (1, 4)$, $(0, 2) \to (-2, 6)$.
- The image has the same side lengths: the bottom side is still 3 long.
Answer: $(-2, 4)$, $(1, 4)$, $(-2, 6)$
Common mistake
Changing the wrong coordinate: moving up or down changes $y$, and moving left or right changes $x$. A quick sketch on a grid catches this.