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Symmetric with Respect to the y-axis

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

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

 

Graph with x and y axes showing a wave curve symmetric about the y-axis, with mirror-image humps on both sides.

 

 

See also

Symmetric with respect to the x-axis, symmetric with respect to the origin, even function, odd function

Key Formula

f(x)=f(−x)for all x in the domainf(x) = f(-x) \quad \text{for all } x \text{ in the domain}
Where:
  • f(x)f(x) = The value of the function at x
  • f(−x)f(-x) = The value of the function at the opposite input, −x
  • xx = Any value in the domain of the function

Worked Example

Problem: Determine whether the function f(x) = x⁴ − 3x² + 5 is symmetric with respect to the y-axis.
Step 1: Write down the symmetry test. A graph is symmetric with respect to the y-axis if f(−x) = f(x) for all x.
Test: f(−x)=f(x)?\text{Test: } f(-x) = f(x)?
Step 2: Replace every x in the function with −x.
f(−x)=(−x)4−3(−x)2+5f(-x) = (-x)^4 - 3(-x)^2 + 5
Step 3: Simplify each term. Remember that (−x) raised to an even power equals x raised to that power.
(−x)4=x4,(−x)2=x2(-x)^4 = x^4, \quad (-x)^2 = x^2
Step 4: Combine the simplified terms.
f(−x)=x4−3x2+5f(-x) = x^4 - 3x^2 + 5
Step 5: Compare f(−x) with the original f(x). They are identical, so the function is symmetric with respect to the y-axis.
f(−x)=x4−3x2+5=f(x)✓f(-x) = x^4 - 3x^2 + 5 = f(x) \quad \checkmark
Answer: Yes, f(x) = x⁴ − 3x² + 5 is symmetric with respect to the y-axis because f(−x) = f(x).

Another Example

This example uses an implicit equation rather than an explicit function, showing that the replacement test x → −x works for any type of equation, not just y = f(x).

Problem: Determine whether the equation x² + y⁴ = 16 (which does not define y as a function of x) is symmetric with respect to the y-axis.
Step 1: For an equation (not necessarily a function), replace x with −x throughout and see if the equation is unchanged.
(−x)2+y4=16(-x)^2 + y^4 = 16
Step 2: Simplify. Since (−x)² = x², the equation becomes:
x2+y4=16x^2 + y^4 = 16
Step 3: Compare with the original equation x² + y⁴ = 16. They are identical, confirming y-axis symmetry.
x2+y4=16(unchanged)✓x^2 + y^4 = 16 \quad \text{(unchanged)} \quad \checkmark
Answer: Yes, the equation x² + y⁴ = 16 is symmetric with respect to the y-axis.

Frequently Asked Questions

What is the difference between symmetric with respect to the y-axis and an even function?
They describe the same property from two perspectives. An even function satisfies f(−x) = f(x), which means its graph is symmetric with respect to the y-axis. The term 'even function' is used when discussing function properties algebraically, while 'symmetric with respect to the y-axis' is used when describing the geometric appearance of a graph. Every even function has y-axis symmetry, and every function whose graph has y-axis symmetry is even.
How do you test if a graph is symmetric with respect to the y-axis?
Replace every x in the equation with −x and simplify. If the resulting equation is identical to the original, the graph is symmetric with respect to the y-axis. This works because reflecting a point (x, y) across the y-axis gives (−x, y), so both points must satisfy the same equation.
Can a graph be symmetric with respect to both the y-axis and the origin?
Yes, but only in a limited case. If a function is symmetric about both the y-axis and the origin, it must satisfy f(x) = f(−x) and f(−x) = −f(x) simultaneously, which forces f(x) = 0 for all x. However, for non-function relations (like a pair of intersecting lines), both symmetries can coexist nontrivially.

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

Symmetric with Respect to the y-axisSymmetric with Respect to the x-axis
DefinitionGraph is unchanged when reflected across the y-axisGraph is unchanged when reflected across the x-axis
Algebraic testReplace x with −x; equation stays the sameReplace y with −y; equation stays the same
Point relationshipIf (a, b) is on the graph, so is (−a, b)If (a, b) is on the graph, so is (a, −b)
Function connectionEquivalent to being an even functionA graph with x-axis symmetry is generally NOT a function (fails vertical line test), unless y = 0
Common examplesy = x², y = |x|, y = cos xx = y², the circle x² + y² = r²

Why It Matters

Y-axis symmetry appears throughout algebra, precalculus, and calculus. Recognizing it lets you sketch graphs faster — you only need to plot the right half and mirror it. In calculus, knowing a function is even (y-axis symmetric) simplifies definite integrals, because ∫−aaf(x) dx=2∫0af(x) dx\int_{-a}^{a} f(x)\,dx = 2\int_{0}^{a} f(x)\,dx when ff is even.

Common Mistakes

Mistake: Replacing y with −y instead of x with −x when testing for y-axis symmetry.
Correction: Reflecting across the y-axis changes the sign of x, not y. Replace x with −x. Replacing y with −y tests for x-axis symmetry instead.
Mistake: Assuming a function with some matching points like f(2) = f(−2) must be symmetric about the y-axis.
Correction: The condition f(x) = f(−x) must hold for every x in the domain, not just a few selected values. Always verify algebraically by substituting −x for x and simplifying the entire expression.

Related Terms