Calculator
Example data table
| Case | Input | Expected symmetry | Reason |
|---|---|---|---|
| Parabola | y = x² | Y-axis | Even function because f(-x) = f(x). |
| Cubic | y = x³ | Origin | Odd function because f(-x) = -f(x). |
| Diagonal relation | Points on y = x | Line y = x | Swapping x and y preserves each point. |
| Measured data | (-2,4), (2,4) | Y-axis | Mirrored x-values share the same y-value. |
Formula used
This calculator checks symmetry by reflecting the graph or point set and comparing the reflected coordinates against the original data within a tolerance.
- Y-axis symmetry: For functions, test
f(-x) = f(x). For points, reflect(x, y)to(-x, y). - X-axis symmetry: Reflect
(x, y)to(x, -y). A function only passes when the graph remains unchanged after that reflection. - Origin symmetry: For functions, test
f(-x) = -f(x). For points, reflect(x, y)to(-x, -y). - Line y = x symmetry: Reflect
(x, y)to(y, x). - Line y = -x symmetry: Reflect
(x, y)to(-y, -x). - Symmetry score:
(matched tests ÷ tested reflections) × 100.
How to use this calculator
- Select Function if you want to test an expression such as
x^2orsin(x). - Select Points if you want to test a custom coordinate set.
- Enter a tolerance. Smaller values require tighter symmetry matches.
- For functions, set the x-range and sample count.
- Submit the form to show results above the calculator.
- Review the score cards, inspect the graph, and export the summary when needed.
Frequently asked questions
1. What does graph symmetry mean?
Graph symmetry means a graph looks unchanged after a reflection or rotation-based coordinate transformation. Common checks include the x-axis, y-axis, origin, and diagonal lines.
2. Why is y-axis symmetry linked to even functions?
If a function satisfies f(-x) = f(x), opposite x-values create equal y-values. That produces a mirror image across the y-axis.
3. Why is origin symmetry linked to odd functions?
If a function satisfies f(-x) = -f(x), reflecting any point through the origin lands on the same graph. Cubic functions often show this behavior.
4. Why might a nearly symmetric dataset fail?
Rounded numbers, measurement noise, or too few sampled points can break exact reflection matches. Increasing tolerance often fixes practical datasets.
5. Can a function be symmetric about the x-axis?
Usually no, unless the graph is entirely on y = 0. Otherwise one x-value would need two opposite y-values, which breaks the definition of a single-valued function.
6. What does the symmetry score show?
The score shows how many reflected checks matched the original graph or point set. A perfect score means every tested reflection found its partner.
7. When should I raise the tolerance?
Raise tolerance when your data comes from experiments, rounding, or imported coordinates with slight noise. Lower tolerance is better for exact algebraic expressions.
8. What is the difference between function mode and points mode?
Function mode samples a rule like y = x² across a range. Points mode checks only the coordinates you provide, which suits measured or imported datasets.