How do you reflect a function over the x-axis?

Put a minus in front: −f(x). Every height changes sign, so the point (a, b) goes to (a, −b).

How do you reflect a function over the y-axis?

Put a minus on the x: f(−x). Every x changes sign, so the point (a, b) goes to (−a, b).

What does it mean if f(−x) equals f(x)?

The graph is its own mirror image over the y-axis, and the function is called even. The parabola y = x² is an example.

What happens if you reflect over both axes?

The graph turns around the origin. The point (a, b) goes to (−a, −b), which is the function −f(−x).

Part 24 of 30 · TransformationsSee the whole course

Reflecting a graph over the x-axis and the y-axis

By The Pip team · October 24, 2026 · 6 min read

Short answer

A minus in front, −f(x), flips the graph upside down over the x-axis: every height changes sign. A minus on the x, f(−x), flips it left to right over the y-axis: every x changes sign.

Two flips, two places for the minus

You already know the rule from the last posts: outside acts on heights, inside acts on x. A minus follows the same pattern.

−f(x): the minus is outside, so it changes the heights. The point (a, b) goes to (a, −b). The graph flips over the x-axis.
f(−x): the minus is inside, so it changes x. The point (a, b) goes to (−a, b). The graph flips over the y-axis.

Do both, −f(−x), and (a, b) goes to (−a, −b): the graph turns all the way around the origin.

Example 1: a line, f(x) = 2x + 1

−f(x) = −(2x + 1) = −2x − 1. The point (1, 3) goes to (1, −3).
f(−x) = 2(−x) + 1 = −2x + 1. The point (1, 3) goes to (−1, 3).

Notice that the flip over the y-axis changes the slope’s sign but keeps the y-intercept, while the flip over the x-axis changes both.

Example 2: the square root, √x

y = −√x hangs below the x-axis. The point (4, 2) goes to (4, −2), and the range becomes y ≤ 0.
y = √(−x) goes left. The point (4, 2) goes to (−4, 2), and the domain becomes x ≤ 0.

The square root shows both flips clearly because it only has one side. Try it below, then switch parents.

y = −f(x)

rule: y = −√x
point (4, 2) → (4, −2)
domain: x ≥ 0, range: y ≤ 0

The minus in front flips every height over the x-axis. The point (4, 2) lands on (4, −2).

Example 3: x², where one flip does nothing

−x² flips the bowl into an arch. But (−x)² is just x², because squaring removes the minus. The parabola is its own mirror image over the y-axis. A graph like that is called an even function.

Watch it explained in under a minute

The same idea, narrated step by step. Square version for sharing: download the 1:1 video.

The four mistakes to avoid

Mixing up the two minuses. In front flips over the x-axis, on x flips over the y-axis.
Thinking −f(x) moves the graph down. It flips it. A shift down would be f(x) − 3.
Expecting f(−x) to change x². It does not, because x² is symmetric.
Forgetting the domain flips too. For √(−x), x must be 0 or less.

Try it yourself

Tap a question to see the answer.

1. For f(x) = x + 2, write −f(x) and f(−x).

−f(x) = −(x + 2) = −x − 2. f(−x) = −x + 2, which is −x + 2.

2. Where does y = √(−x) start, which way does it go, and does it pass through (−4, 2)?

It starts at (0, 0) and goes left, so its domain is x ≤ 0. At x = −4, √4 = 2, so yes.

3. A graph has the point (3, −2). Where does that point go after −f(x), and after f(−x)?

After −f(x): (3, 2). After f(−x): (−3, −2).

Where this fits in

Shifts move a graph and flips turn it over. The last move reshapes it: stretch and compression.