★ MATH · GRAPHING · FUNCTIONS
The graph of y = x² passes through (2, 4) — every point on the curve is one input-output pair of the function. Type a function and watch it plot on the canvas, with zoom, presets, a trace value and a table of values. This one is the GRAPHING tool — plotting y = f(x) — the scientific-calculator page is the SCIENTIFIC tool — trig, logs and powers for single values — not a graph; the fraction-calculator page is the FRACTION tool — adding and multiplying fractions — not functions; and the square-root-calculator page is the SQUARE ROOT tool — roots of single numbers — not curves. No graphing calculator exists on this site — this one is the first.
Plot y = f(x) on an interactive canvas: type the function, set the x-range, and read values off the curve with the trace. On the defaults it is y = x², and f(2) is 4 — the point (2, 4) is marked in red. The plot samples 101 points across the shown range, so zoom in to check a steep or gappy stretch.
It means the point with x = 2 and y = 4 sits on the curve — put 2 into the function and 4 comes out, because 2² is 4. Every point on a graph is one input-output pair of the function: the x-coordinate is the input, the y-coordinate is the output. The graph of y = x² is a parabola, a U-shape with its lowest point at (0, 0), rising to 100 at x = 10 and at x = -10. Read any point the same way: find its x, and its height is the function at that x.
The curve is drawn from samples, not from the whole function. The plot evaluates the function at 101 points across the shown range and joins them with straight segments. For a smooth function like x² that is plenty — the samples at x = 5 and x = -7 land exactly on 25 and 49. For a very steep or jumpy function the picture between samples can mislead, so zoom in to check a suspicious stretch before you believe it.
Some functions have gaps, and a gap is not a zero. 1/x is undefined at x = 0 — there is no point on the curve there, and the two arms of the hyperbola run away in opposite directions. √x needs x ≥ 0, so the curve simply does not exist left of the y-axis. The calculator leaves a gap where the function is undefined instead of inventing a point, and the table of values shows a dash there.
Degrees and radians are not the same input. The trig presets read x in radians, the way mathematics writes it: sin(x) reaches 1 at x = π/2, about 1.5708, not at x = 90. If you type sin(90) expecting 1, you will get about 0.894 — the sine of 90 radians, not 90 degrees. The presets use π where the maths needs it.
On the defaults: y = x², x from -10 to 10, trace x = 2.
And the other way: x² = 9 has two solutions, x = 3 and x = -3 — the table shows f(x) = 9 on both rows, because the parabola is symmetric. And 2^x = 32 has one solution, x = 5, because 2⁵ is 32.
The defaults are y = x² over x from -10 to 10, with the trace at x = 2 — chosen because the parabola is the first graph most people meet, and (2, 4) is its most quoted point. They are a starting point, not a lesson: type any function the parser knows — x^2, sin(x), 1/x, sqrt(x), 2^x, |x|, log(x) — set the range, and read the trace. The plot samples 101 points, so a function that jumps or spikes between samples can look smoother than it is; zoom in to check. The trace x must be a number, and an empty or unreadable expression is an error, not a blank graph.
These notes repeat in every explainer section because this page is built to be read start to finish: the curve is drawn from 101 samples; some functions have gaps where they are undefined; and trig inputs are in radians.
| Output | Formula | What it means |
|---|---|---|
| Function | input | x^2. |
| Trace x | input | 2. |
| f(2) | f(x) = x^2 | 4. |
| f(2) (6dp unrounded) | decimal, 6 places | 4.000000. |
| Formula | by function | f(x) = x^2. |
| Breakdown | step by step | f(2) = 4. |
| Verdict | in words | The graph of y = x^2 passes through (2, 4). |
| Reverse: x = ±3 solves x^2 = 9 (smallest non-negative x with x^2 of 9 or more) | step | 3: x^2 at x = 3: f(x) = 9. |
| Reverse: x = 5 solves 2^x = 32 (smallest whole x with 2^x of 32 or more) | step | 5: 2^x at x = 5: f(x) = 32. |
| x^2 at x = -3 | same model | x^2 at x = -3: f(x) = 9. |
| x^2 at x = 0 | same model | x^2 at x = 0: f(x) = 0. |
| sin(x) at x = π/2 | same model | sin(x) at x = π/2: f(x) = 1. |
| 1/x at x = 2 | same model | 1/x at x = 2: f(x) = 0.5. |
| sqrt(x) at x = 16 | same model | sqrt(x) at x = 16: f(x) = 4. |
| 2^x at x = 3 | same model | 2^x at x = 3: f(x) = 8. |
| |x| at x = -5 | same model | |x| at x = -5: f(x) = 5. |
| log(x) at x = 100 | same model | log(x) at x = 100: f(x) = 2. |
| Empty expression | same model | Type a function first — try x^2.. |
| Unknown function foo(x) | same model | Unknown function "foo" — use sin, cos, tan, sqrt, abs, log or ln.. |
| 1/x at x = 0 (division by zero) | same model | 1/x is undefined at x = 0 — the graph has a gap there, not a point.. |
| sqrt(x) at x = -4 (square root of a negative) | same model | sqrt(x) is undefined at x = -4 — the graph has a gap there, not a point.. |
| Unbalanced parentheses, x^2 + (x | same model | Unbalanced parentheses — every opening bracket needs a closing one.. |
| Non-numeric trace x | same model | Enter a number for the trace x.. |
The arithmetic is exact on the numbers you type: the parser evaluates the function you wrote, and the only rounding is the six-place display of each value. The plot samples the function at discrete points (101 across the shown range) and joins them — very steep or discontinuous functions can look misleading between samples; zoom in to check. The trace reads the function at exactly the x you typed, so it is the reliable number when the picture is in doubt. The parser knows + - * / ^, parentheses, x, sin, cos, tan, sqrt, abs, log (base 10), ln, e and pi — anything else is an error, and undefined points are gaps, not zeros.
The graph of y = x² passes through (2, 4) — every point on the curve is one input-output pair of the function. x^2 at x = -3 is 9; sin(x) at x = π/2 is 1; 1/x at x = 2 is 0.5; sqrt(x) at x = 16 is 4; 2^x at x = 3 is 8; |x| at x = -5 is 5; log(x) at x = 100 is 2. Over x from -10 to 10 the x^2 curve runs from 0 at x = 0 up to 100 at x = ±10. x = ±3 solves x^2 = 9 (smallest non-negative x with x^2 of 9 or more): 3. x = 5 solves 2^x = 32 (smallest whole x with 2^x of 32 or more): 5. Source: Bright Side Kit, brightoolbox.com.
The point is (2, 4), because 2² is 4. Every point on the curve is one input-output pair: the x-coordinate is the input, the y-coordinate is the output.
Pick x-values, work out y for each one, and mark the (x, y) points — for y = x² the points (0, 0), (1, 1) and (2, 4) are on the curve. Join them with the smooth shape the function implies; the table on this page lists the values for you.
Because 1 divided by 0 is undefined — there is no point on the curve at x = 0. The two arms of the hyperbola run away in opposite directions, and the calculator leaves a gap instead of inventing a point.
sin(π/2) is 1 — the first peak of the sine wave. The trig presets read x in radians, so π/2 (about 1.5708) is where the wave tops out, not 90.
Find the x whose height matches — x² = 9 has two solutions, x = 3 and x = -3, because the parabola is symmetric. And 2^x = 32 has one solution, x = 5, because 2⁵ is 32.
(0, 0) — 0² is 0, and the parabola only climbs from there. Over x from -10 to 10 it reaches 100 at both ends.
Because the plot joins 101 sampled points — a spike between two samples can be missed or smoothed over. Zoom in on the suspicious stretch; the trace value at an exact x is always the reliable number.
+ - * / ^, parentheses, x, sin, cos, tan, sqrt, abs, log (base 10), ln, e and pi. Anything else — for example foo(x) — is an error, and an empty expression asks you to type a function first.