A reflection is a transformation that flips a figure over a line to make a mirror image. That line is the line of reflection: each point and its image are the same distance from the line, on opposite sides. The image is congruent but faces the other way. Across the y-axis, (x, y) goes to (−x, y).
In Spanish
reflexión
Also called
flip, reflection transformation, mirror image, reflection across a line
Reflecting across the y-axis changes the sign of x and keeps y. C is 2 units right of the line, so C′ is 2 units left of it.
Formal definition and notation
A reflection across a line l maps each point P to the point P′ such that l is the perpendicular bisector of PP′; points on l map to themselves. A reflection is a rigid motion: it preserves lengths and angle measures, and it reverses orientation, so clockwise order of vertices becomes counterclockwise.
Symbol
Read it as
Meaning
A′
A prime
the image of point A after the transformation
rx-axis
reflection across the x-axis
a common way to name the reflection in geometry books
(x, y) → (x, −y)
x, y maps to x, negative y
the coordinate rule for a reflection across the x-axis
Key facts
Same distance, opposite side: the line of reflection is exactly halfway between each point and its image.
The segment joining a point and its image is perpendicular to the line of reflection.
Points on the line of reflection do not move.
Size and shape stay the same, so the image is congruent. Orientation flips: a figure labeled clockwise A, B, C reads counterclockwise after the reflection.
A figure has line symmetry when a reflection maps it onto itself, like a heart across its center line.
its image across the x-axis: the y-coordinate changes sign
(−x, y)
its image across the y-axis: the x-coordinate changes sign
(y, x)
its image across the line y = x: the coordinates swap
Across the line y = −x, swap and change both signs: (x, y) → (−y, −x). For other lines, count the distance to the line and go the same distance past it.
Try it: reflect a shape
Interactive
Choose a line of reflection: the x-axis, the y-axis or the line y = x. The readout gives the rule and the image of every vertex. Compare the letter order on the original and the image.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Reflection examples, step by step
Example 1: Reflect a point across the x-axis
Warm-up
Reflect P(3, −5) across the x-axis.
Across the x-axis, keep x and change the sign of y: (x, y) → (x, −y).
P(3, −5) → P′(3, 5).
Check: P is 5 units below the x-axis, and P′ is 5 units above it.
Answer:P′(3, 5)
Example 2: Reflect a triangle across the line y = x
Core
Triangle JKL has J(1, 4), K(3, −2), L(−1, 0). Reflect it across the line y = x.
Across y = x, swap the coordinates: (x, y) → (y, x).
J(1, 4) → J′(4, 1).
K(3, −2) → K′(−2, 3).
L(−1, 0) → L′(0, −1).
Answer:J′(4, 1), K′(−2, 3), L′(0, −1)
Example 3: Reflect across a vertical line that is not an axis
Test-style
Reflect Q(5, 1) across the line x = 2.
The line x = 2 is vertical, so y stays 1.
Q is 5 − 2 = 3 units to the right of the line.
Go 3 units to the left of the line: x = 2 − 3 = −1.
The rule (−x, y) only works for the y-axis; here it would give (−5, 1), which is wrong.
Answer:Q′(−1, 1)
Common mistakes
✗ MistakeReflecting (4, 3) across the x-axis gives (−4, 3).
✓ InsteadAcross the x-axis: (4, −3). (−4, 3) is the reflection across the y-axis.
Reflecting across the x-axis flips the point up or down, so the y-coordinate changes sign. The axis named is the one that stays fixed.
✗ MistakeReflecting (5, 1) across x = 2 with the y-axis rule: (−5, 1).
✓ InsteadCount to the line: 3 units right of x = 2, so the image is 3 units left, (−1, 1).
The sign rules only work for the axes and the lines y = x, y = −x. For any other line, measure the distance to the line.
✗ MistakeSliding the triangle to the other side of the line without flipping it.
✓ InsteadMap each vertex separately to the same distance on the opposite side; the image faces the other way.
A slide is a translation. A reflection reverses orientation, like your right hand looking like a left hand in a mirror.
A rotation turns a figure around a point and keeps its orientation; a reflection flips it over a line. Across the x-axis, (2, 3) → (2, −3); a 180° rotation about the origin gives (−2, −3).
A translation slides every point the same distance in the same direction, with no flip. A reflection moves points by different amounts depending on how far they are from the line.
Transformation is the general word for moving or resizing a figure; a reflection is one kind, along with translations, rotations and dilations.
Quick check: 3 questions
Question 1 of 3
What is the image of (−6, 2) after a reflection across the y-axis?
Show answer and explanation
Answer: B. Across the y-axis, change the sign of x: (6, 2). (−6, −2) reflects across the x-axis, (6, −2) is a 180° rotation, and (2, −6) swaps and changes signs.
Question 2 of 3
Which property does a reflection NOT keep the same?
Show answer and explanation
Answer: C. A reflection is a rigid motion, so lengths, angles and area stay the same. Orientation reverses: vertices that went clockwise go counterclockwise in the image.
Question 3 of 3
Point M(7, 4) is reflected across the line y = 1. What is its image?
Show answer and explanation
Answer: C. M is 4 − 1 = 3 units above y = 1, so the image is 3 units below it: y = 1 − 3 = −2, giving (7, −2). (7, −4) uses the x-axis rule by mistake.
Reflection: frequently asked questions
What is the rule for a reflection across the x-axis?
(x, y) → (x, −y). Keep the x-coordinate and change the sign of the y-coordinate. For example, (3, 4) becomes (3, −4) and (−2, −5) becomes (−2, 5). The point flips straight up or down across the x-axis, landing the same distance from the axis on the other side.
Is a reflection a rigid transformation?
Yes. A reflection keeps every length and every angle the same, so the image is congruent to the original. Rigid transformations are reflections, rotations and translations. A reflection is the one that flips orientation: a shape labeled A, B, C clockwise appears labeled counterclockwise in its image.
How do you reflect across the line y = x?
Swap the coordinates: (x, y) → (y, x). The point (2, 5) becomes (5, 2), and (−3, 1) becomes (1, −3). Points already on the line, such as (4, 4), stay put. For the line y = −x, swap and change both signs: (2, 5) → (−5, −2).
TransformationA transformation moves or changes a figure on the coordinate plane. The four common transformations are translations (slides), reflections (flips), rotations (turns) and dilations (resizing). The original figure is the preimage and the result is the image, labeled with primes such as A′. Only a dilation changes the size.Geometry · Grade 7, Grade 8, Algebra 1Congruent FiguresCongruent figures have exactly the same size and shape. One can be moved onto the other by sliding, flipping or turning, so every pair of matching sides and angles is equal. Two squares with side 5 are congruent; a 3-4-5 triangle and a 6-8-10 triangle are not, because their sizes differ. The symbol is ≅.Geometry · Grade 7, Grade 8LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Perpendicular LinesPerpendicular lines are lines that intersect to form right angles: their intersection makes a 90-degree angle. The symbol ⊥ means is perpendicular to. On a coordinate plane, two perpendicular lines that are not vertical have slopes that multiply to −1, such as 2 and −12.Geometry · Grade 4, Grade 5, Grade 6CongruentCongruent means exactly the same size and shape. Two segments are congruent when they have the same length, two angles when they have the same measure, and two figures when one can be turned, flipped or slid onto the other. The symbol is ≅: AB ≅ CD reads “segment AB is congruent to segment CD.”Geometry · Grade 8, GeometrySymmetrySymmetry is when a shape matches itself after it is flipped or turned. A shape has line symmetry when a line splits it into two mirror-image halves that match exactly when folded; that fold line is the line of symmetry. It has rotational symmetry when it looks the same after a turn of less than 360°.Geometry · Grade 4, Grade 8, GeometryParallel LinesParallel lines are lines in the same plane that never intersect. They stay the same distance apart and point in the same direction, like the rails of a straight railroad track. On a coordinate plane, parallel lines have the same slope but different y-intercepts. The symbol ∥ means is parallel to.Geometry · Grade 4, Grade 5, Grade 6TranslationA translation is a slide: a transformation that moves every point of a figure the same distance in the same direction, without turning or flipping it. The image is congruent to the original. On the coordinate plane, a translation adds the same amounts to every point: (x, y) → (x + a, y + b).Geometry · Grade 8, GeometryRotationA rotation is a transformation that turns a figure around a fixed point, called the center, by a given angle and direction. Every point stays the same distance from the center, so size and shape do not change. About the origin, a 90° counterclockwise turn sends (x, y) to (−y, x).Geometry · Grade 8, GeometryTriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5Y-AxisThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x-axis at the origin, (0, 0). A point’s y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Geometry · Grade 5, Grade 6, Grade 7OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8DilationA dilation is a transformation that enlarges or shrinks a figure by a scale factor k from a fixed center point. Each point moves along the ray from the center to k times its distance. With center at the origin, (x, y) → (kx, ky). The image keeps its shape, so it is similar, not congruent.Geometry · Grade 8, GeometryCoordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7QuadrantA quadrant is one of the four regions that the x-axis and y-axis split the coordinate plane into. They are numbered I, II, III and IV counterclockwise, starting at the top right. The signs of a point’s coordinates tell its quadrant: (+, +) is I, (−, +) is II, (−, −) is III and (+, −) is IV.Geometry · Grade 5, Grade 6, Grade 7Ordered PairAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Geometry · Grade 5, Grade 6, Grade 7Line of SymmetryA line of symmetry is a line that divides a figure into two halves that are mirror images of each other. If you fold the figure along that line, the two halves match exactly. A square has 4 lines of symmetry, a rectangle has 2, and a parallelogram that is not a rectangle has none.Geometry · Grade 4, Grade 8, GeometrySimilar FiguresSimilar figures have the same shape but not necessarily the same size. Corresponding angles match, and corresponding side lengths are proportional: each side of one figure is the same multiple, the scale factor, of the matching side of the other. A scale drawing is similar to the real object.Geometry · Grade 7, Grade 8, Algebra 1