Reflection Across Y-Axis- Complete Guide
What Is Reflection Across the Y-Axis?
Reflection across the y-axis is a transformation that flips a point or shape over the vertical y-axis like it's a mirror placed on that line. Every point on one side gets mirrored to the opposite side at the same distance from the axis.
That's it. No rotation, no resizing. Just a horizontal flip.
The Rule You Must Memorize
When you reflect a point (x, y) across the y-axis, the result is (-x, y). The x-coordinate changes sign. The y-coordinate stays exactly the same.
Formula: (x, y) → (-x, y)
Keep this in your head. Every problem involving y-axis reflection uses this exact rule.
How to Reflect a Single Point
Step 1: Identify the original point's coordinates
Step 2: Change the sign of the x-coordinate
Step 3: Keep the y-coordinate identical
Step 4: Plot the new point
Example: Reflect point A(3, 4) across the y-axis
Original: (3, 4)
Apply the rule: x becomes -3, y stays 4
New point: (-3, 4)
The distance from the y-axis stays the same. Point (3, 4) is 3 units right of the y-axis. Point (-3, 4) is 3 units left of the y-axis. Perfect.
More Examples
- Reflect (5, 2) → (-5, 2)
- Reflect (-4, 7) → (4, 7)
- Reflect (-1, -3) → (1, -3)
- Reflect (0, 5) → (0, 5) — points on the y-axis don't move
How to Reflect a Shape or Graph
When reflecting an entire shape:
1. Identify the vertices or key points
2. Apply the rule to each point individually
3. Connect the new points in the same order
Example: Reflect triangle with vertices A(2, 1), B(5, 3), C(4, 6) across the y-axis
Apply (x, y) → (-x, y) to each vertex:
- A(2, 1) → A'(-2, 1)
- B(5, 3) → B'(-5, 3)
- C(4, 6) → C'(-4, 6)
Plot A', B', C' and connect them. That's your reflected triangle.
What Happens to Equations?
If you have a function y = f(x) and you reflect it across the y-axis, replace every x with -x:
New equation: y = f(-x)
Example: y = 2x + 3 reflected across y-axis
Replace x with -x: y = 2(-x) + 3 = -2x + 3
The graph flips horizontally. A line sloping upward from left to right now slopes downward from left to right.
Y-Axis vs X-Axis vs Origin: The Comparison
Students mix these up constantly. Here's the clear difference:
| Reflection Type | Rule | What Changes |
|---|---|---|
| Across Y-Axis | (x, y) → (-x, y) | Only x changes sign |
| Across X-Axis | (x, y) → (x, -y) | Only y changes sign |
| Across Origin | (x, y) → (-x, -y) | Both change sign |
Y-axis reflection: horizontal flip. X changes sign.
X-axis reflection: vertical flip. Y changes sign.
Origin reflection: 180° rotation. Both change sign.
Common Mistakes to Avoid
Changing both coordinates: No. Only x changes when reflecting across the y-axis.
Confusing y-axis with x-axis: Y-axis is vertical. X-axis is horizontal. The axis you reflect across determines which coordinate changes.
Forgetting points on the axis: Any point with x = 0 sits on the y-axis. It doesn't move when reflected.
Mixing up the origin reflection: Origin reflection changes BOTH signs. That's not the y-axis rule.
Quick Practice
Try these in your head, then check:
- What is the reflection of (-7, 2) across the y-axis? → (7, 2)
- What is the reflection of (0, -5) across the y-axis? → (0, -5)
- Reflect (1, 1), (3, 1), (2, 4) and name the new vertices → (-1, 1), (-3, 1), (-2, 4)
When You'll Actually Use This
Graphing functions, geometry proofs, computer graphics, symmetry problems. That's where y-axis reflections show up in the real world.
If you're working with even and odd functions in algebra, y-axis reflections are fundamental. Even functions are symmetric about the y-axis. That means f(x) = f(-x).
Master this transformation and you'll handle symmetry questions without breaking a sweat.