Reflections
A reflection flips a figure over a line — the mirror line — and produces a mirror image. The image is congruent to the original figure: same size, same shape, just flipped. On the coordinate plane, the three mirror lines you'll see most are the -axis, the -axis, and the line .
The good news is that each of those three reflections has a coordinate rule, so you never have to eyeball the graph. Learn which coordinate changes for each mirror line and reflecting a point becomes a five-second job.
The three reflection rules
Across the -axis: . The -coordinate stays the same and the -coordinate becomes its opposite.
Across the -axis: . Now the -coordinate becomes its opposite and the -coordinate stays the same.
Across the line : . No signs change at all — the two coordinates swap places. In the figure below, the point and its image sit on opposite sides of the dashed mirror line , with their coordinates swapped.
Why the rules work
A reflection puts the image exactly as far from the mirror line as the original point, but on the other side. Take the point and reflect it across the -axis. The point sits units above the axis, so its image must sit units below it — that's . The horizontal position never moved, which is why stays the same and only changes sign.
The graph below shows that pair. The dashed segment crosses the -axis at a right angle, and the axis cuts it exactly in half. Every reflection works this way: the mirror line is the perpendicular bisector of the segment joining a point to its image.
How to keep the rules straight
Students mix up the first two rules constantly, so here is the anchor: the coordinate that matches the axis name is the one that survives. Reflecting across the -axis keeps and flips . Reflecting across the -axis keeps and flips . Think of it as the axis protecting its own letter.
For , don't reach for signs at all — just swap. And if you ever meet , that reflection swaps and flips both signs: .
Worked examples
Example 1: across the x-axis
Reflect the point across the -axis.
Answer:
Example 2: across the y-axis
Reflect the point across the -axis.
Answer:
Example 3: across the line y = x
Reflect the point across the line .
Answer:
Try one yourself
Common questions
Which coordinate changes when I reflect across the x-axis?
The -coordinate. The point moves straight up or down to the other side of the axis, so its horizontal position — the -coordinate — never changes. The rule is .
What happens to a point that sits on the mirror line?
Nothing — it maps to itself. A point like reflected across the -axis is units from the line, so its image is also units from the line, in the same spot.
Is the image the same size as the original?
Yes. A reflection is a rigid motion, so every length and every angle is preserved. The image is congruent to the preimage — the only thing that changes is the orientation, which gets flipped like a mirror image.
How is reflecting across y = -x different from y = x?
Across the coordinates just swap: . Across they swap and both signs flip: . So goes to across , but to across .
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