Reflections

Q: Why does reflecting over the x-axis change the y-coordinate?

Because crossing the $x$-axis means moving vertically, and vertical position is measured by $y$. The point keeps its left-right position ($x$ stays), and its height flips sign.

Q: What happens to a point that is on the axis of reflection?

It stays put. A point like $(4, 0)$ sits on the $x$-axis, so reflecting over the $x$-axis leaves it exactly where it is.

Q: What if a figure is reflected over both axes?

Then both coordinates change sign: $(x, y) \to (-x, -y)$. That combination is the same as rotating the figure 180° about the origin.

A reflection flips a figure over a line, producing a mirror image. In pre-algebra the line is almost always the $x$ -axis or the $y$ -axis, and each axis has its own coordinate rule. The image is the same size and shape as the original — a reflection changes which way the figure faces, not how big it is.

Both rules do the same kind of thing: they change the sign of exactly one coordinate and leave the other alone. The whole skill is knowing which coordinate flips for which axis.

The two reflection rules

Over the $x$ -axis: $(x, y) \to (x, -y)$ . The point moves straight up or down across the horizontal axis, so its $x$ -coordinate stays the same and its $y$ -coordinate changes sign.

Over the $y$ -axis: $(x, y) \to (-x, y)$ . The point moves straight left or right across the vertical axis, so its $y$ -coordinate stays the same and its $x$ -coordinate changes sign.

Notice the pattern is the opposite of what many students guess: reflecting over the $x$ -axis changes the $y$ -coordinate. That is because crossing the $x$ -axis is a vertical move — and vertical position is what $y$ measures.

Telling which axis from a graph

Given a figure and its mirror image, compare one vertex with its matching image vertex and ask: which coordinate changed sign?

If only $y$ changed sign, the reflection is over the $x$ -axis. If only $x$ changed sign, it is over the $y$ -axis. In the graph below, the black triangle's corner at $(1, 1)$ matches the blue corner at $(-1, 1)$ — only the $x$ -coordinate flipped, so this is a reflection over the $y$ -axis.

Reflecting a whole figure

To reflect a triangle or any polygon, apply the rule to each vertex, plot the image points, and connect them. Each image point sits the same distance from the axis as its original, just on the other side — a vertex $3$ units above the $x$ -axis reflects to a vertex $3$ units below it.

Points that sit on the line of reflection do not move at all. Reflecting $(4, 0)$ over the $x$ -axis gives $(4, 0)$ — its $y$ -coordinate is $0$ , and $-0$ is still $0$ .

Worked examples

Example 1: reflect over the x-axis

Find the image of $(5, -3)$ after a reflection over the $x$ -axis.

Write the rule for the

x

-axis

(x, y) \to (x, -y)

Keep the

x

-coordinate

x = 5

Change the sign of the

y

-coordinate

-(-3) = 3

Answer: $(5, 3)$

Example 2: reflect over the y-axis

Find the image of $(-4, 1)$ after a reflection over the $y$ -axis.

Write the rule for the

y

-axis

(x, y) \to (-x, y)

Change the sign of the

x

-coordinate

-(-4) = 4

Keep the

y

-coordinate

y = 1

Answer: $(4, 1)$

Example 3: reflect a triangle over the x-axis

Reflect triangle $ABC$ with $A(1, 2)$ , $B(3, 5)$ , $C(4, 1)$ over the $x$ -axis.

The rule changes the sign of each

y

(x, y) \to (x, -y)

Apply the rule to

A(1, 2)

A' = (1, -2)

Apply the rule to

B(3, 5)

B' = (3, -5)

Apply the rule to

C(4, 1)

C' = (4, -1)

Answer: $A'(1, -2)$ , $B'(3, -5)$ , $C'(4, -1)$

Try one yourself

Common questions

Why does reflecting over the x-axis change the y-coordinate?

Because crossing the $x$ -axis means moving vertically, and vertical position is measured by $y$ . The point keeps its left-right position ( $x$ stays), and its height flips sign.

What happens to a point that is on the axis of reflection?

It stays put. A point like $(4, 0)$ sits on the $x$ -axis, so reflecting over the $x$ -axis leaves it exactly where it is.

Does a reflection change the size of the figure?

No. The image is congruent to the original — same side lengths and angles. Only the orientation flips, like a left hand becoming a right hand in a mirror.

What if a figure is reflected over both axes?

Then both coordinates change sign: $(x, y) \to (-x, -y)$ . That combination is the same as rotating the figure 180° about the origin.

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Reflections

The two reflection rules

Telling which axis from a graph

Reflecting a whole figure

Worked examples

Example 1: reflect over the x-axis

Example 2: reflect over the y-axis

Example 3: reflect a triangle over the x-axis

Try one yourself

Common questions

Keep learning

Translations

Rotations

Dilations