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Reflections of Functions

Simple Explanation

A reflection flips a graph over an axis like a mirror. y = −f(x) flips the graph over the x-axis (upside down); y = f(−x) flips it over the y-axis (left-right mirror image).

Why Do We Need It?

Reflections let you instantly picture a "flipped" version of a known shape — useful for recognizing negative-coefficient variants of familiar functions at a glance.

See It

y = x² and its reflection y = −x²

Formula

Reflecting a Function

y = −f(x) reflects over the x-axis. y = f(−x) reflects over the y-axis.

Negating the whole function output flips the graph upside down (over the x-axis); negating the input flips it left-right (over the y-axis).

f(x): — the original function

When to use it: Whenever a graph needs to be mirrored over an axis.

Worked Example

Write the equation of a reflected function

Write the equation of y = √x reflected over the x-axis.

Why Does This Work?

Negating the output of every point (x, f(x)) moves it to (x, −f(x)) — the same horizontal position, but flipped vertically across the x-axis, which is exactly what a mirror reflection over that axis does to every point at once.

Real-Life Example

Mirrored architectural blueprints

An architect designs a building feature, then needs a mirror-image version for the opposite side of a symmetric building.

Reflecting the design's function over an axis produces the exact mirror-image curve needed, with no need to redesign from scratch.

Practice

What is y = x³ reflected over the y-axis?

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Common mistake

Mixing up which reflection formula flips over which axis — y=−f(x) flips over the x-axis (negate the output); y=f(−x) flips over the y-axis (negate the input).

Quick Review

y = −f(x): reflection over the x-axis (negate the output).
y = f(−x): reflection over the y-axis (negate the input).
Every point (x,y) maps to its mirror image across the chosen axis.