Reflection Calculator
Calculate reflections of points over various lines including x-axis, y-axis, y=x, and custom lines with step-by-step solutions and visual diagrams.
Understanding Reflections
What is a Reflection?
A reflection is a transformation that flips a point or shape over a line called the line of reflection. The reflected image is the same distance from the line of reflection as the original, but on the opposite side.
Common Reflection Formulas
Over x-axis: (x, y) → (x, -y)
Over y-axis: (x, y) → (-x, y)
Over y = x: (x, y) → (y, x)
Over y = -x: (x, y) → (-y, -x)
Over y = k: (x, y) → (x, 2k - y)
Over x = h: (x, y) → (2h - x, y)
Through origin: (x, y) → (-x, -y)
Properties of Reflections
- Preserved: Distance (isometry), angle measures, shape and size
- Changed: Orientation (clockwise ↔ counterclockwise)
- The line of reflection is the perpendicular bisector of the segment connecting each point to its image
- Reflections are their own inverse: reflecting twice returns to the original position
Real-World Applications
- Mirror Images: Understanding how objects appear in mirrors
- Symmetry in Design: Architecture, art, and logo design
- Physics: Light reflection, optics, and ray tracing
- Computer Graphics: Rendering reflections in video games and animations
- Biology: Bilateral symmetry in organisms
Frequently Asked Questions
How do you reflect a point over the x-axis?
To reflect a point over the x-axis, keep the x-coordinate the same and change the sign of the y-coordinate. For example, (3, 4) becomes (3, -4). The point flips vertically across the x-axis.
What's the difference between reflection over y = x and y = -x?
Reflection over y = x swaps the coordinates: (x, y) → (y, x). Reflection over y = -x swaps and negates: (x, y) → (-y, -x). The line y = x has a 45° angle, while y = -x has a -45° angle.
How do you reflect over a horizontal line y = k?
Use the formula (x, y) → (x, 2k - y). Find the distance from the point to the line (|y - k|), then move the same distance on the opposite side. For example, reflecting (2, 5) over y = 3 gives (2, 1).
Is reflection the same as rotation?
No. Reflection flips a shape over a line, changing its orientation. Rotation turns a shape around a point, preserving orientation. However, a 180° rotation about the origin produces the same result as point reflection through the origin.
What happens when you reflect a point twice over the same line?
Reflecting twice over the same line returns the point to its original position. Reflection is its own inverse, meaning R ∘ R = I (identity transformation).
Can you reflect over diagonal lines other than y = x?
Yes! You can reflect over any line. For lines like y = mx + b, use the general formula: find the perpendicular distance to the line and place the reflected point the same distance on the opposite side.
How do reflections preserve distance?
Reflections are isometries (distance-preserving transformations). The distance between any two points equals the distance between their reflected images. This property makes reflections useful in geometry and physics.
What is point reflection?
Point reflection (or central reflection) reflects through a point rather than a line. For reflection through the origin: (x, y) → (-x, -y). This is equivalent to a 180° rotation about that point.