Reflections

Learn how to reflect shapes across lines of symmetry and understand mirror images in geometry.

Intermediate25 minLesson
Printable worksheet

Definition

A reflection is a transformation that flips a figure over a line, creating a mirror image. The line is called the line of reflection or line of symmetry.
Key properties of reflections:
  • The original figure (pre-image) and the reflected figure (image) are congruent (same size and shape)
  • Each point and its reflection are equidistant from the line of reflection
  • The line of reflection is the perpendicular bisector of the segment connecting any point to its image
Common lines of reflection:
  • x-axis: (x,y)→(x,−y)(x, y) \rightarrow (x, -y)
  • y-axis: (x,y)→(−x,y)(x, y) \rightarrow (-x, y)
  • **Line y=xy = x**: (x,y)→(y,x)(x, y) \rightarrow (y, x)

Try it now

What is the rule for reflecting a point across the x-axis?

Worked Examples

Reflect the point A(3,5)A(3, 5) across the x-axis.
1

Identify the line of reflection

The line of reflection is the x-axis (the horizontal line y=0y = 0) → Line: x-axis
2

Apply the reflection rule

For reflection across the x-axis: (x,y)→(x,−y)(x, y) \rightarrow (x, -y) → Keep x, negate y
3

Calculate the new coordinates

A(3,5)→A′(3,−5)A(3, 5) \rightarrow A'(3, -5)
4

Verify the reflection

Point AA is 5 units above the x-axis. Point A′A' is 5 units below the x-axis. Both are equidistant from the line of reflection. → Verified

Common Mistakes

Negating the wrong coordinate when reflecting

Why it's wrong: Students confuse which coordinate changes for each axis. For x-axis reflection, y changes. For y-axis reflection, x changes.

Correct: Remember: reflect across x-axis changes y (vertical flip), reflect across y-axis changes x (horizontal flip)

Forgetting the negative sign when reflecting across axes

Why it's wrong: Students keep the original sign instead of negating the appropriate coordinate.

Correct: Always check: if a point is above the x-axis, its reflection is below (negative y), and vice versa

Thinking reflection changes the size of the shape

Why it's wrong: Students confuse reflection with other transformations like dilation.

Correct: Reflections preserve size and shape. The image is always congruent to the pre-image.

Confusing the order of coordinates when reflecting across y = x

Why it's wrong: Students may not realize that this reflection swaps the x and y values.

Correct: For y=xy = x reflection: just swap the coordinates. (3,7)(3, 7) becomes (7,3)(7, 3)

Interactive Visual

Transformation Grid

T(3, 2)
-10-10-8-8-6-6-4-4-2-2224466881010xy(1, 3)(4, 3)(2, 6)(4, 5)(7, 5)(5, 8)
OriginalTransformed

Function graph

-7-6-5-4-3-2-11234567-2-1123456780xyCC'PQRP'Q'R'

Tap or hover over the graph to read off values.

y = x

Practice Problems

16 problems
Problem 1 of 16
Easy
What is the rule for reflecting a point across the x-axis?

Why It Matters

Reflections appear everywhere in the world around us:
  • Art and Design: Artists use reflections to create symmetrical patterns, logos, and balanced compositions
  • Architecture: Buildings often feature symmetrical facades using reflection principles
  • Nature: Butterflies, leaves, and faces exhibit approximate reflectional symmetry
  • Physics: Light reflects off mirrors following the same geometric principles
  • Computer Graphics: Video games and animations use reflections to create realistic water surfaces and mirrors

Real World Applications

Mirror Reflections

When you look in a mirror, you see a reflection of yourself. Your left hand appears as your right hand in the mirror image.

Example:

If you stand 2 feet from a mirror, your reflection appears 2 feet behind the mirror surface, for a total apparent distance of 4 feet.

1Try It Yourself

A point P(6,4)P(6, 4) represents where you are standing. The mirror is along the y-axis.

Where does your reflection appear?

Step 1: Write the mathematical expression

Apply the y-axis reflection rule:

Symmetric Logo Design

Many company logos use reflectional symmetry to create visually appealing and balanced designs.

Example:

The McDonald's 'M' has vertical symmetry - if you reflect the left half across a vertical line, you get the right half.

2Try It Yourself

A designer places a point at (3,5)(3, 5) on the right side of a logo. The line of symmetry is the y-axis.

Where should the matching point be on the left side?

Step 1: Write the mathematical expression

Reflect (3,5)(3, 5) across the y-axis:

Kaleidoscope Patterns

Kaleidoscopes create beautiful patterns by reflecting objects across multiple lines, producing symmetrical designs.

Example:

A simple kaleidoscope uses two mirrors at an angle to reflect a small collection of colored beads into a complex pattern.

3Try It Yourself

A colored bead is at position (2,3)(2, 3). The kaleidoscope first reflects it across the x-axis, then across the y-axis.

Where is the final position of the bead image?

Step 1: Write the mathematical expression

Apply two reflections in sequence:

Key Takeaways

  • 1A reflection flips a figure over a line of reflection, creating a mirror image
  • 2For x-axis reflection: (x,y)→(x,−y)(x, y) \rightarrow (x, -y) - negate the y-coordinate
  • 3For y-axis reflection: (x,y)→(−x,y)(x, y) \rightarrow (-x, y) - negate the x-coordinate
  • 4For reflection across y=xy = x: (x,y)→(y,x)(x, y) \rightarrow (y, x) - swap the coordinates
  • 5Reflections preserve size and shape - the image is congruent to the pre-image
  • 6Each point and its image are equidistant from the line of reflection

Frequently Asked Questions

Think about which direction you're flipping. Reflecting across a horizontal line (like the x-axis) flips vertically, so y changes. Reflecting across a vertical line (like the y-axis) flips horizontally, so x changes.
Think about which direction you're flipping. Reflecting across a horizontal line (like the x-axis) flips vertically, so y changes. Reflecting across a vertical line (like the y-axis) flips horizontally, so x changes.
The point stays in the same place! If a point is on the line of reflection, it is its own image. For example, (0,5)(0, 5) reflected across the y-axis is still (0,5)(0, 5).
No. A reflection creates a mirror image (like flipping), while a rotation turns the figure around a point. However, two reflections can sometimes produce the same result as a rotation.

Glossary

Reflection
A transformation that flips a figure over a line, creating a mirror image
Line of reflection
The line over which a figure is reflected; also called the line of symmetry
Pre-image
The original figure before a transformation is applied
Image
The resulting figure after a transformation is applied
Congruent
Having the same size and shape
Perpendicular bisector
A line that cuts a segment in half at a 90-degree angle
Today's challengeOne problem, three tries

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