Symmetric with Respect to the x-axis

Q: Can a function be symmetric with respect to the x-axis?

No — with the sole exception of $y = 0$. If a graph is symmetric about the x-axis, then for any point $(x, y)$ the point $(x, -y)$ is also on the graph. That means one $x$-value maps to two different $y$-values (one positive, one negative), which violates the definition of a function. So graphs with x-axis symmetry are relations, not functions.

Q: How do you algebraically test for x-axis symmetry?

Replace every $y$ in the equation with $-y$ and simplify. If the resulting equation is equivalent to the original, the graph is symmetric with respect to the x-axis. If it is not equivalent, the graph does not have this symmetry.

Symmetric about the x-axis
Symmetric across the x-axis
Symmetric with Respect to the x-axis

Describes a graph that is left unchanged when reflected across the x-axis.

A curve symmetric about the x-axis on an x-y graph, showing two mirrored branches reflected across the x-axis.

Key Formula

\text{If } (x,\, y) \text{ is on the graph, then } (x,\, -y) \text{ is also on the graph.}

Where:

$(x, y)$ = Any point on the graph
$(x, -y)$ = The reflection of that point across the x-axis; the x-coordinate stays the same while the y-coordinate changes sign

Worked Example

Problem: Test whether the equation

x = y^2

is symmetric with respect to the x-axis.

Step 1: Write down the original equation.

x = y^2

Step 2: Replace every

y

with

-y

in the equation.

x = (-y)^2

Step 3: Simplify the right side. Since squaring a negative gives a positive,

(-y)^2 = y^2

x = y^2

Step 4: Compare the result to the original equation. They are identical, so the graph is symmetric with respect to the x-axis.

Answer: Yes, the graph of

x = y^2

is symmetric with respect to the x-axis. For every point like

(4, 2)

on the parabola, the reflected point

(4, -2)

is also on the parabola.

Another Example

This example shows a case that fails the test, helping students see what a negative result looks like. It also involves a common function (a cubic) that students might mistakenly assume has x-axis symmetry.

Problem: Test whether the equation

y = x^3

is symmetric with respect to the x-axis.

Step 1: Write down the original equation.

y = x^3

Step 2: Replace

y

with

-y

-y = x^3

Step 3: Solve for

y

to compare more easily.

y = -x^3

Step 4: Compare

y = -x^3

to the original

y = x^3

. These are not equivalent equations, so the graph is NOT symmetric with respect to the x-axis.

Answer: No, the graph of

y = x^3

is not symmetric with respect to the x-axis. For example,

(2, 8)

is on the graph but

(2, -8)

is not.

Frequently Asked Questions

Can a function be symmetric with respect to the x-axis?

No — with the sole exception of

y = 0

. If a graph is symmetric about the x-axis, then for any point

(x, y)

the point

(x, -y)

is also on the graph. That means one

x

-value maps to two different

y

-values (one positive, one negative), which violates the definition of a function. So graphs with x-axis symmetry are relations, not functions.

How do you algebraically test for x-axis symmetry?

Replace every

y

in the equation with

-y

and simplify. If the resulting equation is equivalent to the original, the graph is symmetric with respect to the x-axis. If it is not equivalent, the graph does not have this symmetry.

What is the difference between x-axis symmetry and y-axis symmetry?

X-axis symmetry means the top and bottom halves of the graph are mirror images (replace

y

with

-y

). Y-axis symmetry means the left and right halves are mirror images (replace

x

with

-x

). A graph can have one, both, or neither type of symmetry.

Symmetric with Respect to the x-axis vs. Symmetric with Respect to the y-axis

	Symmetric with Respect to the x-axis	Symmetric with Respect to the y-axis
Reflection axis	Reflects across the horizontal x-axis	Reflects across the vertical y-axis
Algebraic test	Replace y with −y; equation unchanged	Replace x with −x; equation unchanged
Point condition	If (x, y) is on graph, so is (x, −y)	If (x, y) is on graph, so is (−x, y)
Can it be a function?	Generally no (fails vertical line test)	Yes — these are called even functions
Classic example	x = y² (sideways parabola)	y = x² (standard parabola)

Why It Matters

You encounter x-axis symmetry when graphing conic sections such as ellipses (

x^2/a^2 + y^2/b^2 = 1

) and hyperbolas, which are symmetric about both axes. Recognizing this symmetry lets you plot only the top half of a curve and reflect it downward, cutting your graphing work in half. It also appears in physics and engineering when analyzing shapes like parabolic reflectors and satellite dishes described by equations of the form

x = ay^2

Common Mistakes

Mistake: Confusing x-axis symmetry with y-axis symmetry and replacing x with −x instead of y with −y.

Correction: For x-axis symmetry, the reflection flips points vertically, so you replace y with −y. For y-axis symmetry, the reflection flips points horizontally, so you replace x with −x. Think about which coordinate changes sign when you reflect across each axis.

Mistake: Assuming a graph with x-axis symmetry can be a function.

Correction: If both

(x, y)

and

(x, -y)

are on the graph (with

y \neq 0

), that single

x

-value has two outputs, which violates the definition of a function. Graphs symmetric about the x-axis (other than

y = 0

) are relations, not functions.

Related Terms

Symmetric with Respect to the y-axis — Mirror image across the vertical axis instead
Symmetric with Respect to the Origin — 180° rotational symmetry about the origin
Even Function — Functions with y-axis symmetry, f(−x) = f(x)
Odd Function — Functions with origin symmetry, f(−x) = −f(x)
Graph of an Equation or Inequality — The visual representation being tested for symmetry
Reflection — The geometric transformation underlying symmetry
Relation — X-axis symmetric graphs are relations, not functions