Symmetric With Respect To The Origin - Mathwords

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Symmetric about the Origin Symmetric across the Origin Symmetric with Respect to the Origin

Describes a graph that looks the same upside down or right side up. Formally, a graph is symmetric with respect to the origin if it is unchanged when reflected across both the x-axis and y-axis.

See also

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

Key Formula

If (x, y) is on the graph, then (−x, −y) is also on the graph.Algebraic test: Replace x with −x and y with −y. If the equation is unchanged, the graph is symmetric with respect to the origin.f(−x)=−f(x)(for functions)\begin{gathered}\text{If } (x,\, y) \text{ is on the graph, then } (-x,\, -y) \text{ is also on the graph.}\\\text{Algebraic test: Replace } x \text{ with } {-x} \text{ and } y \text{ with } {-y}. \text{ If the equation is unchanged, the graph is symmetric with respect to the origin.}\\f(-x) = -f(x) \quad \text{(for functions)}\end{gathered}If (x,y) is on the graph, then (−x,−y) is also on the graph.Algebraic test: Replace x with −x and y with −y. If the equation is unchanged, the graph is symmetric with respect to the origin.f(−x)=−f(x)(for functions)​Where:

• (x,y)(x, y)(x,y) = Any point on the original graph
• (−x,−y)(-x, -y)(−x,−y) = The point obtained by negating both coordinates, equivalent to a 180° rotation about the origin
• f(x)f(x)f(x) = A function whose origin symmetry is being tested

Worked Example

Problem: Determine whether the graph of y = x³ is symmetric with respect to the origin.Step 1: Replace x with −x and y with −y in the equation.(−y)=(−x)3(-y) = (-x)^3(−y)=(−x)3Step 2: Simplify the right side. A negative number cubed stays negative.−y=−x3-y = -x^3−y=−x3Step 3: Multiply both sides by −1 to solve for y.y=x3y = x^3y=x3Step 4: Compare the result to the original equation. The original equation was y = x³, and we arrived back at y = x³. The equation is unchanged, so the graph is symmetric with respect to the origin.Answer: Yes, y = x³ is symmetric with respect to the origin.

Another Example

This example uses a relation (not a function) and illustrates that origin symmetry is not exclusive to odd functions. It also shows a case where the graph has all three types of symmetry at once.

Problem: Determine whether the equation x² + y² = 25 (a circle centered at the origin) is symmetric with respect to the origin.Step 1: Replace x with −x and y with −y in the equation.(−x)2+(−y)2=25(-x)^2 + (-y)^2 = 25(−x)2+(−y)2=25Step 2: Simplify. Squaring a negative number gives a positive result.x2+y2=25x^2 + y^2 = 25x2+y2=25Step 3: The resulting equation is identical to the original, so the graph is symmetric with respect to the origin.Answer: Yes, x² + y² = 25 is symmetric with respect to the origin. Note that this circle is also symmetric with respect to the x-axis and y-axis — a graph can have multiple types of symmetry simultaneously.

Frequently Asked Questions

What is the difference between symmetric with respect to the origin and symmetric with respect to the y-axis?A graph symmetric with respect to the y-axis satisfies f(−x) = f(x): the left and right halves are mirror images across the vertical axis. A graph symmetric with respect to the origin satisfies f(−x) = −f(x): rotating the graph 180° about the origin leaves it unchanged. For example, y = x² is symmetric about the y-axis, while y = x³ is symmetric about the origin.Are all odd functions symmetric with respect to the origin?Yes. By definition, an odd function satisfies f(−x) = −f(x) for all x in its domain, which is exactly the algebraic condition for origin symmetry. Conversely, if a function's graph is symmetric with respect to the origin, that function is odd.How do you test for origin symmetry by plugging in points?Pick any point (a, b) that satisfies the equation, then check whether (−a, −b) also satisfies it. For example, if (2, 8) lies on y = x³, check (−2, −8): (−8) = (−2)³ = −8 ✓. Testing a few points can build intuition, but the algebraic substitution test is the only way to prove symmetry for the entire graph.

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

Symmetric with Respect to the Origin	Symmetric with Respect to the y-axis
Definition	Graph unchanged after 180° rotation about the origin	Graph unchanged when reflected across the y-axis
Algebraic test	Replace x with −x and y with −y; equation unchanged	Replace x with −x; equation unchanged
For functions	f(−x) = −f(x) (odd function)	f(−x) = f(x) (even function)
Classic example	y = x³, y = sin x, y = 1/x	y = x², y = cos x, y = |x|
Visual cue	Opposite quadrants mirror each other (Q1↔Q3, Q2↔Q4)	Left half is a mirror image of the right half

Why It Matters

Origin symmetry appears throughout algebra, precalculus, and calculus whenever you classify functions as odd or even. Recognizing it lets you sketch graphs faster — you only need to plot one half and rotate. In calculus, knowing a function is odd tells you immediately that its definite integral over any interval symmetric about zero equals zero, which saves significant computation.

Common Mistakes

Mistake: Only replacing x with −x (but not y with −y) when testing for origin symmetry.Correction: You must replace both variables. Replacing only x tests for y-axis symmetry, not origin symmetry. For functions written as y = f(x), an equivalent shortcut is checking whether f(−x) = −f(x).Mistake: Assuming a graph that passes through the origin must be symmetric with respect to the origin.Correction: Passing through (0, 0) is not sufficient. For example, y = x² + x passes through the origin but is not symmetric about it, since f(−x) = x² − x ≠ −(x² + x). You must verify the algebraic condition for all x, not just one point.

Related Terms

• Symmetric with Respect to the x-axis — Reflection across the x-axis only
• Symmetric with Respect to the y-axis — Reflection across the y-axis only
• Odd Function — Functions with origin symmetry satisfy f(−x) = −f(x)
• Even Function — Functions with y-axis symmetry, the natural counterpart
• Graph of an Equation or Inequality — The visual representation being tested for symmetry
• Rotation — Origin symmetry is a 180° rotation about (0, 0)
• Reflection — Origin symmetry equals reflections across both axes

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