Symmetric
Symmetric
Describes a geometric figure or a graph consisting of two parts that are congruent to each other.
See also
Axis of symmetry, point of symmetry, symmetric with respect to the origin, symmetric with respect to the x-axis, symmetric with respect to the y-axis
Key Formula
Symmetric w.r.t. the y-axis: f(x)=f(−x)for all x
Symmetric w.r.t. the origin: f(−x)=−f(x)for all x
Symmetric w.r.t. the x-axis: if (x,y) is on the graph, then (x,−y) is also on the graph
Where:
- f(x) = The function or equation being tested for symmetry
- x = Any input value in the domain
- y = The output or vertical coordinate
- −x = The opposite of the input, used to test reflection across the y-axis or origin
- −y = The opposite of the output, used to test reflection across the x-axis
Worked Example
Problem: Determine whether the graph of f(x) = x² + 4 is symmetric with respect to the y-axis.
Step 1: Write the symmetry test for the y-axis. A graph is symmetric about the y-axis if f(-x) = f(x) for every x.
Test: f(−x)=f(x)?
Step 2: Compute f(-x) by replacing every x in the formula with -x.
f(−x)=(−x)2+4=x2+4
Step 3: Compare f(-x) with the original f(x).
f(−x)=x2+4=f(x)
Step 4: Since f(-x) equals f(x) for all x, the condition is satisfied.
Answer: Yes, f(x) = x² + 4 is symmetric with respect to the y-axis. Its graph is a parabola opening upward, and the y-axis is its axis of symmetry.
Another Example
This example differs from the first because it tests an equation (not a function) for all three types of symmetry. A circle is a rare case where all three hold simultaneously. Note that this relation is not a function because it fails the vertical line test, yet x-axis symmetry still applies.
Problem: Determine all types of symmetry for the equation x² + y² = 25.
Step 1: Test for y-axis symmetry by replacing x with -x and checking whether the equation is unchanged.
(−x)2+y2=x2+y2=25✓
Step 2: Test for x-axis symmetry by replacing y with -y.
x2+(−y)2=x2+y2=25✓
Step 3: Test for origin symmetry by replacing both x with -x and y with -y.
(−x)2+(−y)2=x2+y2=25✓
Step 4: All three substitutions produce the original equation, so the graph has all three types of symmetry.
Answer: The circle x² + y² = 25 is symmetric with respect to the y-axis, the x-axis, and the origin.
Frequently Asked Questions
How do you test if a graph is symmetric?
You use algebraic substitution. Replace x with -x to test y-axis symmetry, replace y with -y to test x-axis symmetry, or replace both x with -x and y with -y to test origin symmetry. If the resulting equation is equivalent to the original, that type of symmetry exists.
What is the difference between symmetric about the y-axis and symmetric about the origin?
Y-axis symmetry means the left half of the graph mirrors the right half across the y-axis; algebraically, f(-x) = f(x). These are called even functions. Origin symmetry means rotating the graph 180° about the origin leaves it unchanged; algebraically, f(-x) = -f(x). These are called odd functions. A graph cannot be both unless it is the constant function f(x) = 0.
Can a function be symmetric about the x-axis?
No. If a graph is symmetric about the x-axis, then for some x-value there would be two y-values (y and -y), which violates the definition of a function. Only relations that are not functions—such as circles or horizontal parabolas like x = y²—can have x-axis symmetry.
Y-axis symmetry (even function) vs. Origin symmetry (odd function)
| Y-axis symmetry (even function) | Origin symmetry (odd function) | |
|---|---|---|
| Algebraic test | f(-x) = f(x) | f(-x) = -f(x) |
| Geometric meaning | Left side mirrors right side across the y-axis | Graph looks the same after a 180° rotation about the origin |
| Classic examples | x², cos(x), |x| | x³, sin(x), 1/x |
| Graph behavior at origin | Not required to pass through origin | Must pass through the origin (if defined there) |
| Alternate name | Even function | Odd function |
Why It Matters
Symmetry simplifies graphing because you only need to plot half the graph and reflect. In calculus, knowing a function is even lets you write ∫−aaf(x)dx=2∫0af(x)dx, while knowing it is odd means the integral over a symmetric interval equals zero. Symmetry also appears throughout physics, engineering, and art whenever balanced or mirror-image structures arise.
Common Mistakes
Mistake: Assuming that because a graph looks symmetric, it must satisfy the algebraic test.
Correction: Always verify symmetry algebraically. For example, y = x² + x looks nearly symmetric near its vertex but fails the test f(-x) ≠ f(x), so it is not symmetric about the y-axis.
Mistake: Confusing origin symmetry with y-axis symmetry when testing f(-x).
Correction: After computing f(-x), compare it to both f(x) and -f(x). If f(-x) = f(x), you have y-axis symmetry. If f(-x) = -f(x), you have origin symmetry. These are two different outcomes of the same substitution.
Related Terms
- Axis of Symmetry — The line a symmetric figure reflects across
- Point of Symmetry — The center point for rotational symmetry
- Symmetric with Respect to the y-axis — Specific case where f(-x) = f(x)
- Symmetric with Respect to the x-axis — Reflection symmetry across the x-axis
- Symmetric with Respect to the Origin — 180° rotational symmetry, f(-x) = -f(x)
- Congruent — The two halves of a symmetric figure are congruent
- Geometric Figure — Shapes that can exhibit symmetry
- Graph of an Equation or Inequality — Graphs are tested for symmetry algebraically