A vertical reflection flips a graph or figure upside down across the x-axis, like a mirror lying flat on the ground. Every point above the x-axis moves to the same distance below it, and every point below moves above. It’s one of the most common transformations you’ll encounter in algebra and geometry, and the rule behind it is straightforward: the x-coordinate stays the same while the y-coordinate switches sign.
How a Vertical Reflection Works
A reflection is a transformation that creates a mirror image of a shape or graph by flipping it over a line. In a vertical reflection, that line is the x-axis. Think of it as folding your graph paper along the horizontal axis. Any point at (x, y) lands on (x, −y). So the point (3, 5) becomes (3, −5), and the point (−2, −4) becomes (−2, 4). The horizontal position never changes. Only the vertical position flips.
This makes a vertical reflection a purely vertical transformation. It doesn’t shift anything left or right, stretch anything wider, or rotate anything. It simply inverts the height of every point on the figure.
The Algebraic Rule
If you’re working with functions, the rule is even simpler. Given a function f(x), its vertical reflection is the new function g(x) = −f(x). You multiply all the outputs by −1. That’s it.
For example, if f(x) = x², then its vertical reflection is g(x) = −x². The original parabola opens upward; the reflected version opens downward. Every y-value that was positive becomes negative, and vice versa. The shape itself is identical, just flipped. If f(3) = 9, then g(3) = −9. The curve still passes through the origin because flipping zero gives you zero.
This works for any function. If you have a line, a cubic, an exponential curve, or anything else, placing a negative sign in front of the entire function reflects it vertically across the x-axis.
Vertical vs. Horizontal Reflection
The naming here trips people up, so it’s worth being precise. A vertical reflection flips the graph vertically (up becomes down) across the x-axis. A horizontal reflection flips the graph horizontally (left becomes right) across the y-axis. The direction of the flip and the axis it crosses are different things.
The algebraic rules are distinct too:
- Vertical reflection: g(x) = −f(x). You negate the output. The coordinate change is (x, y) → (x, −y).
- Horizontal reflection: g(x) = f(−x). You negate the input. The coordinate change is (x, y) → (−x, y).
A quick way to remember: vertical reflection puts the negative sign outside the function (affecting y-values), while horizontal reflection puts it inside (affecting x-values).
Shape and Size Stay the Same
A vertical reflection is a rigid transformation, also called an isometry. That means it doesn’t change the size or shape of the figure. Distances between points remain identical, angles stay the same, and the reflected image is congruent to the original. You could cut the shape out of paper, flip it over, and it would match perfectly.
This is true of all reflections, not just vertical ones. Translations (slides) and rotations are also rigid transformations. What sets reflections apart is that they reverse orientation. If the original figure has points labeled clockwise, the reflected figure will have them counterclockwise. The shape is preserved, but the “handedness” flips, the same way your left hand becomes a right hand in a mirror.
Applying It to Coordinates
When you need to reflect a set of points vertically, apply the rule (x, y) → (x, −y) to each one. Suppose you have a triangle with vertices at (1, 2), (4, 6), and (5, 1). After a vertical reflection, those vertices become (1, −2), (4, −6), and (5, −1). Plot the new points, connect them, and you have the reflected triangle sitting below the x-axis as a mirror image of the original.
Points that sit directly on the x-axis don’t move at all, since their y-coordinate is already zero and negating zero changes nothing. These points act as anchor points where the original figure and its reflection touch.
If you need to reflect across a horizontal line other than the x-axis, the process requires an extra step. You first calculate each point’s vertical distance from that line, then place the reflected point the same distance on the opposite side. But when people refer to a vertical reflection without specifying a line, they almost always mean a reflection across the x-axis.