A graph shows how one input changes another quantity across a coordinate system, including repeating or curved behavior.
Use it to see rate, shape, turning points, period, phase, or domain behavior rather than treating a result as a one-off number.
The relationship
Write the rule before calculating
(x−h)²=4p(y−k) or (y−k)²=4p(x−h).
See the structure
What the calculation is doing
∿ ∿ ∿one period / curve(x−h)²=4p(y−k) or (y−k)²=4p(x−h).
Worked interpretation
Read the result in context
Vertex (0,0), p=1 has focus (0,1).
Interpret with care
Important boundary
A graph’s scale, domain, period, and phase conventions matter; a drawn curve is not proof of global behavior.
Use the Mathematics collection to move between connected concepts without duplicating the calculation.
Quick guide
How to use this calculator
Select a supported direction or geometric form when modes are available.
Enter the labelled coordinates, coefficients, distances, or transformation parameters.
Read the primary result, then inspect classification, equation, closest-point, or graph details.
Calculation method
Apply the coordinate relationship
(x−h)²=4p(y−k) or (y−k)²=4p(x−h).
The calculator validates degenerate points, lines, planes, conics, coordinate-system domains, and bounded graph sampling before reporting a result.
Worked example
Worked example
Vertex (0,0), p=1 has focus (0,1).
(x−h)²=4p(y−k) or (y−k)²=4p(x−h).
Supported inputs
Precision and limits
Coordinate conventions
Cartesian coordinates use right-handed x, y, z axes. Polar and cylindrical azimuth is measured counterclockwise from positive x; spherical φ is the polar angle from positive z.
Numerical range
Inputs may be zero or have magnitude from 1e-100 through 1e100. Trigonometric angles are limited to ±1e9°. Fixed-step plots limit Lissajous frequencies to ±20 and polar frequencies to ±30, preserving at least 12 samples per cycle. Generated grids and plots use explicit point caps.
Precision and graphs
Calculations use double-precision arithmetic and normally display 12 significant digits. Graphs are bounded numerical visualizations, not proofs of global behavior.
