Solve
Type a function of x —or an equality— and the graph appears as you type.
Loading tool…
Quick guide
Understand the calculation
A visual tool to graph and analyze up to two mathematical functions simultaneously, exploring their spatial properties in real-time. It marks roots, maxima, minima, inflection points and the value at x = 0, and computes the intersections when two curves are present.
When to use this tool
Ideal for studying the analytical behavior of a function, visually identifying its roots (x-intercepts), finding maximum or minimum points, understanding where multiple equations intersect, and analyzing graphical trends.
How it works
Type in your mathematical expression (e.g., x^2 - 4) or set up an equality (like cos(x) = x); in that second case the grapher plots the difference between both sides, so the roots of the curve are the solutions of the equation. As you enter the formula or drag across the cartesian plane, the grapher instantly plots the path and automatically highlights its intersections, roots, and key points. The whole analysis is numerical and is redone over the range you are looking at: the grapher splits that range into 500 intervals, looks for sign changes —of the function for roots, of its derivative for extrema— and refines each candidate by bisection. Use your mouse wheel to zoom or simply drag the canvas to explore freely; zooming in is not only a larger view, it also concentrates the sampling over a narrower span.
The analysis is numerical and lives inside the visible range: with 500 sampling intervals, a feature narrower than that step can go unnoticed until you zoom in. Roots are detected by sign change, so one that touches the axis without crossing it —the 0 of x^2 or of abs(x)— is only found when a sample lands exactly on it; otherwise it shows up as an extremum but not as a root. The lists show up to 10 roots, 6 extrema and 10 intersections; the ∞ symbol means there are more inside the view. It accepts two functions of a single variable x, up to 300 characters each. Functions with vertical asymptotes are drawn interrupted and the tool says so. Nothing is stored: reloading the page discards the expressions.
Examples
- A parabola and its notable points
x^2 - 4 → roots at x = −2 and x = 2; minimum at (0, −4) - An equation with no simple algebraic solution
cos(x) = x → single root at x ≈ 0.7391 - Two curves and where they cross
f = x^2 · g = x + 2 → (−1, 1) and (2, 4)
Frequently asked questions
Can I graph an equation instead of a function?
Yes. When you type something like cos(x) = x, the grapher plots the difference between both sides and marks its roots: each one is a solution of the equation. That is why, in that mode, the value at x = 0 is labelled x=0 rather than f(0): it represents the difference, not the function.
Why does it miss a root I know exists?
Because it looks for sign changes across 500 samples of the visible range. If the curve touches the axis without crossing it —like abs(x) or x^2 at the origin— there is no sign change, and the root only appears when a sample lands right on it: that is why it can show up and then vanish as you zoom. The extremum, on the other hand, is always marked. If the crossing does exist but is narrower than the sampling step, zooming in fixes it, because the same number of samples covers a narrower span.
What does the ∞ symbol at the end of a list mean?
That there are more roots, extrema or intersections inside the visible range than the list can show. This is common for periodic functions such as sin(x), which have infinitely many roots. It also appears when both functions coincide across the whole range, where listing intersection points would make no sense.
Are the functions kept when I reload the page?
No. Everything lives in the tab's memory: reloading or closing it discards the expressions and the view.