A reflection flips a figure across a line, like a mirror image. It is a rigid motion, like a translation: the size and shape don’t change.
Why it works
Think of the mirror line as a fold. Each point goes straight across the line and lands the same distance away on the other side. Across the $x$-axis (a horizontal line), points move straight up or down, so $x$ stays and $y$ changes sign. Across the $y$-axis (a vertical line), points move straight left or right, so $y$ stays and $x$ changes sign.
Key idea
Across the $x$-axis: $(x, y) \to (x, -y)$. Across the $y$-axis: $(x, y) \to (-x, y)$. Across the line $y = x$: $(x, y) \to (y, x)$.
Worked examples
Worked example: Across the x-axis
Reflect $(3, -5)$ across the $x$-axis.
- The point is 5 below the axis, so its image is 5 above it.
- $x$ stays 3.
Answer: $(3, 5)$
Worked example: Across the y-axis
Reflect $(3, -5)$ across the $y$-axis.
- The point is 3 to the right of the axis, so its image is 3 to the left.
- $y$ stays $-5$.
Answer: $(-3, -5)$
Worked example: Across another line
Reflect $(-1, 4)$ across the vertical line $x = 2$.
- The point is 3 units left of $x = 2$.
- The image is 3 units right of it: $x = 5$. The height stays 4.
Answer: $(5, 4)$
Common mistake
Reflecting across the $y$-axis by changing the sign of $y$. The $y$-axis is vertical, so the point flips left to right and its $x$ changes sign: $(3, -5) \to (-3, -5)$. Changing $y$ instead would be a reflection across the $x$-axis. A quick sketch shows which way the point should move.
A reflection reverses the order of the corners: if $A$, $B$, $C$ go around clockwise, their images go around counterclockwise. Lengths and angles still stay the same.