Functions Reflected by the Y-Axis- Graphing Guide
What Y-Axis Reflection Actually Means
When you reflect a function across the y-axis, you're creating a mirror image along the vertical line x = 0. Every point on the original graph gets flipped to the opposite side of the y-axis, while the y-coordinate stays exactly the same.
The transformation rule is simple: (x, y) → (-x, y). Take any point, negate the x-value, keep the y-value, and you have the reflected point.
That's it. No complicated math. Just flipping horizontally.
Why This Matters for Graphing
Y-axis reflections show up constantly in algebra and calculus. They're the visual representation of an even function — a function where f(x) = f(-x) for every x in the domain.
When you see y = f(-x) instead of y = f(x), you're looking at a reflection. The entire graph flips across the y-axis. This isn't optional knowledge if you're working with functions.
Quick Visual Check
Still confused? Try this mental test: imagine the y-axis is a mirror. If you held a graph up to that mirror, where would points appear? That's your reflection.
- Point at (3, 4) becomes (-3, 4)
- Point at (-5, 2) becomes (5, 2)
- Point at (0, 7) stays at (0, 7) — the y-axis is the line of symmetry
Y-Axis Reflection vs. X-Axis Reflection
Don't confuse these two. They're completely different operations.
| Reflection Type | Transformation | What Changes |
|---|---|---|
| Y-Axis | (x, y) → (-x, y) | Only the x-coordinate flips |
| X-Axis | (x, y) → (x, -y) | Only the y-coordinate flips |
The y-axis reflection keeps y values intact. The x-axis reflection keeps x values intact. Mix these up and you'll graph complete nonsense.
Common Function Types and Their Y-Axis Reflections
Linear Functions
Take y = 2x + 3. Reflect it across the y-axis and you get y = -2x + 3. The slope changes sign. The y-intercept stays the same because it's on the y-axis itself.
Quadratic Functions
The parabola y = x² is already symmetric across the y-axis — it's an even function. Reflecting it does nothing visible because it's already its own mirror image.
But y = (x - 2)²? That's different. Reflecting gives y = (-x - 2)², which simplifies to y = (x + 2)². The vertex shifts from (2, 0) to (-2, 0).
Absolute Value Functions
The graph of y = |x| is V-shaped and symmetric. Reflecting y = |x - 3| across the y-axis gives y = |-x - 3|, which simplifies to y = |x + 3|. The vertex moves from (3, 0) to (-3, 0).
Exponential Functions
y = 2ˣ grows to the right. Reflecting across the y-axis gives y = 2⁻ˣ, which is the same as y = (1/2)ˣ — a decaying exponential that falls to the left instead.
How to Graph a Y-Axis Reflection: Step by Step
Here's the actual process for reflecting any function across the y-axis.
Method 1: Point-by-Point
- Identify key points on the original graph (intercepts, vertices, turning points)
- Change each x-coordinate to its opposite (multiply by -1)
- Keep the y-coordinates the same
- Plot the new points and connect them the same way the original was connected
Method 2: Using the Function Rule
If you have y = f(x) and you want to graph y = f(-x), just substitute -x everywhere x appears in the original function.
Example: Starting with f(x) = x³ - 4x
To find f(-x), substitute: (-x)³ - 4(-x) = -x³ + 4x
So the reflected function is y = -x³ + 4x.
Practical Example: Reflecting y = √x
The function y = √x only exists for x ≥ 0. Its graph starts at (0, 0) and curves upward to the right.
Reflecting across the y-axis gives y = √(-x). Now the function only exists for x ≤ 0. The graph starts at (0, 0) and curves upward to the left.
Result: two mirror-image curves on opposite sides of the y-axis.
Common Mistakes to Avoid
- Neglecting the domain: After reflection, the domain flips. A function defined only for x ≥ 0 now exists only for x ≤ 0.
- Reflecting the y-coordinate: Don't change y. Only x gets negated.
- Forgetting about the y-intercept: Points on the y-axis (where x = 0) don't move during y-axis reflection.
- Confusing the notation: y = f(-x) is a reflection. y = -f(x) is an x-axis reflection. These are not the same.
Y-Axis Reflection in Parent Functions
| Original Function | Y-Axis Reflection | Key Change |
|---|---|---|
| y = x² | y = (-x)² = x² | No visible change (already symmetric) |
| y = x³ | y = (-x)³ = -x³ | Graph flips upside down |
| y = 1/x | y = 1/(-x) = -1/x | Graph flips upside down |
| y = |x| | y = |-x| = |x| | No visible change (already symmetric) |
| y = √x | y = √(-x) | Entire graph moves to left side |
Notice that even functions (x², |x|) look identical after reflection. Odd functions (x³, 1/x) flip orientation.
When You'll Actually Use This
Y-axis reflections aren't just textbook exercises. They come up in:
- Transformations of parent functions — the foundation of graphing higher-level functions
- Even and odd function classification — determining symmetry properties
- Graph sketching from equations — quickly generating related graphs
- Physics applications — mirror images, symmetry in systems
The Bottom Line
Y-axis reflection is basic transformation geometry. Negate x, keep y, plot the result. The concept is straightforward. The only reason people struggle is overcomplicating it or confusing it with x-axis reflection.
If you can't sketch a reflection after reading this, go back and trace a few points by hand. That's what actually builds the intuition.