What's the difference between y = -f(x) and y = f(-x)?

y = -f(x) flips the graph across the x-axis (top-to-bottom flip). y = f(-x) flips it across the y-axis (left-to-right flip). The position of the negative sign tells you which axis: if the negative is outside the function (on the output), it's the x-axis; if it's inside (on the input), it's the y-axis.

How do I check if I reflected correctly?

Pick a point on the original graph and see where it moves. If the original graph has the point (2, 3) and you reflect across the x-axis, the new graph must have (2, -3). If you reflect across the y-axis, it must have (-2, 3). Test another point to be sure.

What happens if I need to reflect across both axes at once?

Use the formula y = -f(-x), which applies both transformations. This is equivalent to rotating the graph 180° around the origin. Points (x, y) become (-x, -y). If a problem asks for only one reflection, don't use this formula.

Will reflecting a graph a second time undo the first reflection?

Yes, but only if you reflect across the same axis twice. Reflecting across the x-axis twice returns the graph to its original position. Reflecting across the x-axis and then the y-axis is different—that's the same as rotating 180°.

Reflecting a Graph

Reflecting a graph flips it across an axis—use it when you need to reverse a function's behavior, either top-to-bottom or left-to-right.

y = -f(x) \text{ or } y = f(-x)

Solve a problem with Reflecting a Graph

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What each symbol means

What Reflecting a Graph takes

Symbol	Meaning
$f$	The function name representing a rule that produces output values; if you read it as just a label instead of a function producing $y = f(x)$ , you'll miss that you're transforming the outputs themselves.
$x$	The input value (independent variable) plugged into the function; confusing it with $y$ makes you negate the wrong part and reflect across the wrong axis.
$y$	The output value (dependent variable) produced by $f(x)$ ; it represents the $y$ -coordinates of points on the graph, and reflecting across the x-axis means negating these outputs.

When to use it

Use this when you need to flip a function's graph across the x-axis or y-axis to model the opposite behavior of a relationship.

Level

Usually taught in: Algebra II

Worked examples

1. Reflecting a Linear Function Across the x-Axis

Problem

f(x) = 2x - 3

, write the equation of the graph after reflecting it across the x-axis.

$f(x) = 2x - 3$
We start with the original function. For any input $x$ , the output is $2x - 3$ .
$y = -f(x)$
To reflect across the x-axis, use the formula $y = -f(x)$ , which negates every output value.
$y = -(2x - 3)$
Substitute the original function into the reflection formula. Now we apply the negative sign to the entire expression.
$y = -2x + 3$
Distribute the negative sign carefully: $-(2x) = -2x$ and $-(-3) = +3$ .

Answer: $y = -2x + 3$

Reflecting across the x-axis flips every point upside down. Points that were above the x-axis move below it, and their $y$ -coordinates become negatives. By negating the entire output $f(x)$ , we reverse the sign of every $y$ -coordinate on the graph while keeping all $x$ -coordinates the same.

2. Reflecting a Quadratic Function Across the y-Axis

Problem

Given

f(x) = x^2 - 4x + 1

, find the equation after reflecting across the y-axis.

$f(x) = x^{2} - 4x + 1$
We start with a quadratic function. This parabola opens upward because the coefficient of $x^{2}$ is positive.
$y = f(-x)$
To reflect across the y-axis, use the formula $y = f(-x)$ . This means replace every $x$ in the original function with $-x$ .
$y = (-x)^{2} - 4(-x) + 1$
Substitute $-x$ everywhere $x$ appears: the $x^{2}$ becomes $(-x)^{2}$ , and the $-4x$ term becomes $-4(-x)$ . Be careful with signs.
$y = x^{2} + 4x + 1$
Simplify each term: $(-x)^{2} = x^{2}$ (negative squared is positive), and $-4(-x) = 4x$ (negative times negative is positive).

Answer: $y = x^{2} + 4x + 1$

Reflecting across the y-axis mirrors the graph left-to-right. The parabola's shape stays the same, but it's flipped horizontally. Notice that $x^{2}$ stays the same under reflection (since $(-x)^{2} = x^{2}$ ), but the $x$ term changes sign from $-4x$ to $4x$ . This is the key: linear terms always flip sign when you replace $x$ with $-x$ .

3. Reflecting a Quadratic Relationship in a Physics Context

Problem

A ball's height above ground is modeled by

h(t) = -5t^{2} + 20t

, where

t

is time in seconds. To represent the opposite scenario (like if gravity pulled upward instead of downward), write the equation of the graph reflected across the x-axis.

$h(t) = -5t^{2} + 20t$
The original function models realistic falling motion. The negative coefficient on $t^{2}$ means the parabola opens downward.
$y = -h(t)$
To reflect across the horizontal axis (x-axis), use $y = -h(t)$ , which negates every height value.
$y = -(-5t^{2} + 20t)$
Substitute the original function. The negative sign applies to the entire expression, so we distribute it across all terms.
$y = 5t^{2} - 20t$
Distribute the negative: $-(-5t^{2}) = 5t^{2}$ and $-(20t) = -20t$ . The parabola now opens upward.

Answer: $y = 5t^{2} - 20t$

This reflection flips the trajectory upside down, representing reversed gravity. The original function showed a ball rising then falling (parabola opening down). The reflected function shows the opposite pattern (parabola opening up), as if height values were negated. All the same time values apply, but their corresponding heights are opposite, turning a descending arc into an ascending one.

Common mistakes

Where Reflecting a Graph usually goes wrong

The mistake	Why it is wrong	The fix
Writing $y = 3x - 2$ (unchanged) when reflecting $f(x) = 3x - 2$ across the x-axis.	They forgot to apply the reflection formula $y = -f(x)$ , so the function didn't transform at all.	Use $y = -f(x) = -(3x - 2) = -3x + 2$ to negate every output value.
Writing $y = f(-x) = (-x)^{2} = x^{2}$ when reflecting $f(x) = x^{2}$ across the x-axis.	The formula $f(-x)$ reflects across the y-axis (input negation), not the x-axis (output negation). In this case, $x^{2}$ is symmetric across the y-axis, so the reflection looks unchanged, hiding the error.	For x-axis reflection, use $y = -f(x) = -x^{2}$ , which flips the parabola upside down.
Writing $y = -f(-x) = -(-x)^{2} = -x^{2}$ when asked to reflect $f(x) = x^{2}$ across only the x-axis.	Applying both transformations (negating both input and output) creates a 180° rotation around the origin, not a single reflection across one axis.	For a single reflection, use only one transformation: $y = -f(x) = -x^{2}$ for x-axis, or $y = f(-x) = x^{2}$ for y-axis reflection.

Tips and when to use something else

When reflecting across the x-axis, every $y$ -value becomes its opposite: the point $(x, y)$ moves to $(x, -y)$ . This is what $y = -f(x)$ accomplishes.
When reflecting across the y-axis, every $x$ -value becomes its opposite: the point $(x, y)$ moves to $(-x, y)$ . This is what $y = f(-x)$ accomplishes.
If a graph has special features like intercepts or turning points, reflecting will flip those too. An x-intercept stays an x-intercept, but y-intercepts move to their opposites.
Use Translating a Graph if you need to shift the function left, right, up, or down—translations add or subtract, not negate. Reflections are flips; translations are slides.

Related formulas

Translating a GraphTranslate graphs using y = f(x - h) + k to shift any function horizontally by h and vertically by k units, needed when repositioning known curves.Equation of a ParabolaThe equation (x - h)^2 = 4p(y - k) describes a parabola with vertex at (h, k) and tells you which direction it opens and how wide it is.Absolute ValueAbsolute value measures the distance a number is from zero without regard to direction, so it is always positive or zero.

Frequently asked questions

What's the difference between $y = -f(x)$ and $y = f(-x)$ ?: $y = -f(x)$ flips the graph across the x-axis (top-to-bottom flip). $y = f(-x)$ flips it across the y-axis (left-to-right flip). The position of the negative sign tells you which axis: if the negative is outside the function (on the output), it's the x-axis; if it's inside (on the input), it's the y-axis.
How do I check if I reflected correctly?: Pick a point on the original graph and see where it moves. If the original graph has the point $(2, 3)$ and you reflect across the x-axis, the new graph must have $(2, -3)$ . If you reflect across the y-axis, it must have $(-2, 3)$ . Test another point to be sure.
What happens if I need to reflect across both axes at once?: Use the formula $y = -f(-x)$ , which applies both transformations. This is equivalent to rotating the graph 180° around the origin. Points $(x, y)$ become $(-x, -y)$ . If a problem asks for only one reflection, don't use this formula.
Will reflecting a graph a second time undo the first reflection?: Yes, but only if you reflect across the same axis twice. Reflecting across the x-axis twice returns the graph to its original position. Reflecting across the x-axis and then the y-axis is different—that's the same as rotating 180°.

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Reviewed 2026-09-18