Reflections
A reflection flips a figure over a line, like a mirror image. Every point lands the same distance from the line, on the other side.
Know these first
- Translations 8.G.1
- Master List skill 6.NS.6
This unlocks
- Rotations 8.G.1
- Congruence Through Rigid Motions 8.G.2
What you'll be able to do
- Reflect points and figures over the x-axis, the y-axis and the line y = x.
- Reflect over vertical and horizontal lines like x = 3 or y = −1.
- Find the mirror line from a point and its image.
- Combine two reflections and describe the single move that matches.
The big idea
The mirror line sits exactly halfway between each point and its image, and the segment joining them crosses the mirror at 90°.
Over the x-axis, x stays and y flips: (x, y) → (x, −y). Over the y-axis, y stays and x flips: (x, y) → (−x, y). Over y = x, the coordinates swap: (x, y) → (y, x).
For a line like x = 3, count: a point 4 units left of the mirror lands 4 units right of it. A reflection keeps lengths and angles but reverses orientation - clockwise corners become counterclockwise.
Words you need
- Line of reflection
- The mirror line a figure is flipped over.
- Image
- Where a point lands after the flip, often written with a prime: A′.
- Orientation
- The clockwise or counterclockwise order of a figure's corners. Reflections reverse it.
Reflection rules
| Mirror line | Rule | Example |
|---|---|---|
| x-axis | (x, y) → (x, −y) | (3, 2) → (3, −2) |
| y-axis | (x, y) → (−x, y) | (3, 2) → (−3, 2) |
| y = x | (x, y) → (y, x) | (3, 2) → (2, 3) |
| y = −x | (x, y) → (−y, −x) | (3, 2) → (−2, −3) |
| x = k | (x, y) → (2k − x, y) | over x = 2: (5, 1) → (−1, 1) |
Worked examples
Example 1
Reflect the point (3, 2) over the x-axis.
- Over the x-axis, x stays the same and y changes sign.
- (3, 2) → (3, −2).
Answer: (3, −2)
Example 2
Reflect triangle ABC with A(1, 2), B(4, 2) and C(1, 5) over the y-axis.
- Over the y-axis, y stays the same and x changes sign.
- A′(−1, 2), B′(−4, 2), C′(−1, 5).
Answer: (−1, 2), (−4, 2), (−1, 5)
Example 3
Reflect the point (5, 1) over the line x = 2.
- The point is 5 − 2 = 3 units right of the mirror.
- Its image is 3 units left of the mirror: x = 2 − 3 = −1.
- y doesn't change, so the image is (−1, 1).
Answer: (−1, 1)
Common mistakes
Changing the wrong coordinate. Over the x-axis, the y-value flips (the point jumps above or below). Over the y-axis, the x-value flips.
Reflecting over x = 3 as if it were the y-axis. Measure the distance to the mirror line itself, then go the same distance past it.
Forgetting that reflections reverse orientation. A reflected figure is a mirror image: its corners run the other way around.
Practice: 20 questions
12 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Reflect the point (4, −3) over the x-axis.
- Reflect the point (−2, 5) over the y-axis.
- A mirror lies along the x-axis. A lamp sits at (2, 4). Where is its reflection? Word problem
- A sheet of paper is folded along the y-axis. A dot of paint at (−3, 1) presses onto the other side. Where does it land? Word problem
- A calm lake's surface is the x-axis. The top of a tree is at (5, 6). Where is the top of its reflection in the water? Word problem
Tier 2 Intermediate One more step to think through
- Reflect triangle ABC with A(1, 1), B(3, 1) and C(2, 4) over the x-axis.
- Reflect the point (2, 5) over the line y = x.
- Point (3, −4) maps to (−3, −4), and point (1, 2) maps to (−1, 2). Which reflection is this?
- A kite in a design sits at (−4, −1). The design is reflected over the line y = x. Where does the kite end up? Word problem
- A butterfly design has left-wing points at (−3, 2) and (−1, 4). The right wing is the reflection over the y-axis. Find the right-wing points. Word problem
- A ball at (6, −2) bounces off the x-axis as a mirror. Where is its reflection? Word problem
- A segment runs from (0, 0) to (6, 8). It is reflected over the y-axis. How long is the image? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- Reflect the point (6, 3) over the line x = 2.
- Reflect the point (−1, 4) over the line y = 1.
- A pond's edge runs along the line x = 3. A duck at (7, 2) sees its reflection. Where is the reflection? Word problem
- The point (2, 3) is reflected over the x-axis, and then the image is reflected over the y-axis. Where does it end up? Word problem
- A sign's corner at (1, 5) appears at (7, 5) in a vertical mirror. Which line is the mirror? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- Reflect the point (3, 1) over the line y = −x.
- The point (1, 2) is reflected over the line x = 3, and then the image is reflected over the line x = 6. Where does it end up? Word problem
- A figure is reflected over two parallel mirrors that are 3 units apart. What single move does the same thing? Word problem
Answers
Tier 1 Foundations
- (4, 3). Apply (x, y) → (x, −y) to each point.
- (2, 5). Apply (x, y) → (−x, y) to each point.
- (2, −4). Apply (x, y) → (x, −y) to each point.
- (3, 1). Apply (x, y) → (−x, y) to each point.
- (5, −6). Apply (x, y) → (x, −y) to each point.
Tier 2 Intermediate
- (1, −1), (3, −1), (2, −4). Apply (x, y) → (x, −y) to each point.
- (5, 2). Apply (x, y) → (y, x) to each point.
- (x, y) → (−x, y). Apply the rule to every original point; each one lands on its image point.
- (−1, −4). Apply (x, y) → (y, x) to each point.
- (3, 2), (1, 4). Apply (x, y) → (−x, y) to each point.
- (6, 2). Apply (x, y) → (x, −y) to each point.
- 10. Across 6, up or down 8: d² = 6² + 8² = 100, so d = 10.
Tier 3 Advanced
- (−2, 3). Apply reflect over x=2 to each point.
- (−1, −2). Apply reflect over y=1 to each point.
- (−1, 2). Apply reflect over x=3 to each point.
- (−2, −3). Apply (x, y) → (x, −y); then (x, y) → (−x, y) to each point.
- reflect over x=4. Apply the rule to every original point; each one lands on its image point.
Tier 4 Challenge
- (−1, −3). Apply (x, y) → (−y, −x) to each point.
- (7, 2). Apply reflect over x=3; then reflect over x=6 to each point.
- A translation of 6 units, at right angles to the mirrors. The first flip and the second flip undo each other's reversal, and the figure ends up shifted by twice the distance between the mirrors.
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