Enter Function Details
Formula Used
| Symmetry Type |
Formula |
Meaning |
| Even function |
f(-x) = f(x) |
The graph has y-axis symmetry. |
| Odd function |
f(-x) = -f(x) |
The graph has origin symmetry. |
| Vertical axis |
f(2h - x) = f(x) |
The graph reflects across the line x = h. |
| Point symmetry |
f(x) + f(2h - x) = 2k |
The graph turns around the center point (h, k). |
| Inverse symmetry |
f(f(x)) = x |
The function behaves like its own inverse. |
How to Use This Calculator
- Enter a function using
x as the variable.
- Choose the x-range for numerical testing.
- Set the number of sample points.
- Enter a tolerance for decimal comparison.
- Add a custom vertical axis or center point if needed.
- Press the calculate button.
- Review the symmetry verdicts, graph, and sample table.
- Use the CSV or PDF buttons to save the result.
Example Data Table
| Function |
Expected Symmetry |
Reason |
x^2 |
Even |
f(-x) equals f(x). |
x^3 |
Odd |
f(-x) equals -f(x). |
sin(x) |
Odd |
The sine curve has origin symmetry. |
cos(x) |
Even |
The cosine curve has y-axis symmetry. |
x^2 + 2*x + 1 |
Vertical axis x = -1 |
The parabola reflects around its vertex line. |
Understanding Function Symmetry
Function symmetry shows how a graph behaves after reflection or rotation.
It helps students read structure before doing long calculations.
A symmetric graph often has predictable values.
This makes checking, sketching, and solving easier.
Even and Odd Patterns
An even function gives the same output for opposite inputs.
Its graph reflects across the y-axis.
Common examples include x^2 and cos(x).
An odd function changes sign when the input changes sign.
Its graph has origin symmetry.
Examples include x^3 and sin(x).
Custom Axes and Centers
Some functions are not symmetric about the standard axes.
A shifted parabola may reflect around x equals h.
A shifted cubic may rotate around a point.
This calculator includes custom axis and center tests.
That makes it useful for transformed graphs.
Numerical Testing
The tool evaluates many points in the selected range.
It compares transformed values with the original values.
A tolerance is used because decimals can contain rounding noise.
A very small tolerance is strict.
A larger tolerance is more forgiving.
Choose a wide range when you need stronger evidence.
Graph and Exports
The graph compares f(x) and f(-x).
The table shows direct values.
These outputs make errors easier to spot.
CSV export is helpful for spreadsheets.
PDF export is useful for records, homework, and reports.
The verdict should be treated as a numerical guide.
For exact proof, simplify the formulas algebraically.
FAQs
1. What is function symmetry?
Function symmetry describes whether a graph matches itself after reflection or rotation. Common checks include y-axis symmetry, origin symmetry, custom vertical-axis symmetry, and point symmetry around a chosen center.
2. What is an even function?
An even function satisfies f(-x) = f(x). Its left and right sides match across the y-axis. Examples include x^2, x^4, and cos(x).
3. What is an odd function?
An odd function satisfies f(-x) = -f(x). Its graph has origin symmetry. If you rotate it 180 degrees around the origin, it matches itself.
4. Can a function have x-axis symmetry?
A normal function usually cannot have x-axis symmetry unless it is always zero. Otherwise, one input would need two opposite output values, which fails the vertical line rule.
5. What does tolerance mean?
Tolerance is the allowed difference between two compared values. It handles rounding errors from decimal calculations. Lower tolerance gives stricter results. Higher tolerance gives more flexible results.
6. Does this calculator prove symmetry?
It gives strong numerical evidence by checking many sample points. It does not replace a formal algebraic proof. Use the formula section to confirm exact symmetry.
7. Which functions are supported?
You can use powers, arithmetic, constants, and common functions. Supported examples include sin, cos, tan, sqrt, abs, log, and exp.
8. Why do some values show undefined?
A value becomes undefined when the expression is outside its domain. Examples include division by zero, square roots of negative values, and logarithms of nonpositive values.