Geometry / Unit 4: Transformations
4.1 Translations
Notes
A transformation moves a figure (the preimage) to a new figure (the image). Image points get primes: $A$ maps to $A'$.
- A translation slides every point the same distance in the same direction.
- Rule: $(x, y) \to (x + a, y + b)$. Right/up is positive; left/down is negative.
- A translation is a rigid motion (isometry): it keeps lengths and angle measures.
- Example: $(x, y) \to (x - 3, y + 5)$ sends $A(2, 1)$ to $A'(-1, 6)$.
Practice
The point $P(4, -2)$ is translated by $(x, y) \to (x + 3, y - 6)$. Where is $P'$?
Add 3 to $x$ and subtract 6 from $y$.
A triangle is translated 5 units left and 2 units up. Which rule describes this?
Left changes $x$ by $-5$; up changes $y$ by $+2$.
$A(1, 3)$ maps to $A'(-2, 7)$ under a translation. Where does $B(5, 0)$ map?
The rule is $(x, y) \to (x - 3, y + 4)$, so $B' = (2, 4)$.
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