This simulator graphs a function of two variables, z = f(x, y), as a colored surface over a [−2, 2]² domain, then slices it with two perpendicular planes — one holding x fixed, one holding y fixed — to show that both traces always meet at exactly one height. Switch between four function families, rescale and shift the surface, and sweep the slicing planes to see how peaks, sign changes and symmetry appear in the cross-sections.
• A real-time 3D view of the colored function-graph mesh, an orange x = a slicing plane with its trace curve z = f(a, y), a violet y = b slicing plane with its trace curve z = f(x, b), a bright probe marking where both traces meet at (a, b, f(a, b)), and two detached cross-section profile ribbons for comparing traces away from the surface — with home view, focus-selected-part, auto-rotate, expand and toggleable labels. • Five live controls: a function-family selector (sin x cos y "waves", xy/2 "saddle", exp(−(x²+y²)/2) "bell", or √(x²+y²)/2 "cone"), vertical scale (0.2-1.5), vertical offset (−1 to 1), slice position x = a and slice position y = b (each −2 to 2). • One action: sweep/animate the slicing planes, plus pause. • Six live metrics: height f(a,b) at the probe, height f(a,0) along the x-trace, height f(0,b) along the y-trace, the base height f(0,0), the current vertical scale and the current vertical offset. • A Curves & measurements tab plotting height f(a,b) and the x-trace height, the full model equations and snapshot measurements. • An Experiments tab with four guided scenarios (saddle sign change, bell symmetry, shifting the cone, scanning the waves), a model-verification bench of independent automated checks, and a timestamped event log with a copyable trial report. • A Learn & assess tab with guided lessons, a two-question knowledge-check quiz and a written scope/reference statement.
For z = f(x, y), a vertical line drawn over any single (x, y) point in the domain meets the graph exactly once — that is what makes it a function graph rather than an arbitrary surface. This is also why a full sphere cannot be represented as one such function over its complete projection: at most points inside its equatorial circle, a vertical line would need to cross the sphere twice, once on the top half and once on the bottom.
Holding x constant at a and letting y vary traces a curve confined to a vertical plane, given by z = f(a, y); holding y constant at b instead traces z = f(x, b). Both curves are cross-sections of the very same surface, so they necessarily agree at the single point where both conditions hold simultaneously — (a, b, f(a, b)), marked by the bright probe.
The four available surfaces behave differently under slicing: the waves surface (sin x cos y) oscillates and can flip sign repeatedly across the domain; the saddle (xy/2) changes sign moving diagonally away from either axis and produces straight-line traces; the bell (a Gaussian) decays symmetrically away from its center peak; and the cone (√(x²+y²)/2) has a sharp, non-smooth apex at the origin despite still being a well-defined single-valued function.
The equations panel shows z = s·g(x, y) + c for the base function g scaled by s and shifted by offset c, with the x = a trace given by z = s·g(a, y) + c and the y = b trace by z = s·g(x, b) + c. Multiplying by the vertical scale stretches height differences throughout the surface, while adding the vertical offset raises or lowers the entire graph uniformly — the offset never alters the surface's horizontal shape, only where it sits vertically, exactly as the shift-the-cone experiment demonstrates by raising the cone's apex without changing its sharpness.
A function graph can only have one height value for each (x, y) point in its domain — a vertical line must cross it exactly once. A sphere fails this test over most of its projected domain, since a vertical line through an interior point of the projected circle crosses the sphere twice, once on the upper hemisphere and once on the lower hemisphere.
No. The offset adds the same constant height to every point on the surface, which raises or lowers the entire graph without changing its shape when viewed from above — peaks stay in the same (x, y) locations, sign changes happen at the same lines, and the two slicing traces still meet at the same (a, b) location, just at a shifted height.
Both traces are cross-sections of the same underlying surface: the x = a trace shows z = f(a, y) for varying y, and the y = b trace shows z = f(x, b) for varying x. Since both come from the identical function f, they necessarily agree at the single point (a, b) where both conditions hold at once, giving the shared height f(a, b) marked by the probe.
This model covers single-valued graphs of two-variable functions sampled over a [−2, 2]² display domain; reported heights at the probe and traces are evaluated analytically rather than read off the sampled mesh. The detached cross-section ribbons are translated display copies of the slicing-plane traces for easier comparison — they are not additional surfaces, volumes, or independent measurements of anything beyond the two live traces.