What this calculator offers
- Parses expressions safely with a Shunting-yard evaluator and plots up to three functions in different colors.
- Detects x-intercepts and intersections by scanning for sign changes, then converging with bisection.
- Identifies extrema via numeric derivatives: f′ brackets the critical point, f″ classifies minima and maxima.
- Encodes inputs, range, and options in the URL so you can share the same view and reproduce the full step log.
For educational use only. Verify the formulas and ranges before relying on the results.
Set up your functions and range
- Pick a preset (or type your own functions).
- Zoom/pan in the graph preview to explore.
- Use “Detected points” to jump to intercepts, intersections, and extrema.
Graph preview
Move the pointer over the canvas to inspect coordinates.
Keyboard shortcuts: arrows pan, +/- zoom, F fits the Y range, R resets the viewport.
Detected points
Points detected: 0
| Sample: sin(x) crosses zero near x = 0; JavaScript lists detected roots, extrema, and intersections. |
How it's calculated
- Parse each expression and sample the selected x-range.
- Draw visible functions on the canvas and look for sign changes or turning points.
- Refine detected points before adding them to the results table.
Teacher notes
- The step log records each bracket, interval width, and g(x) value so students can follow the bisection workflow.
- Extrema rely on f′ sign changes and the sign of f″, making it easy to connect the numeric method to calculus concepts.
- Canvas controls work with mouse, touch, and keyboard, ensuring the same graph can be reproduced in class or online.
How to use this calculator effectively
Use the graphing calculator to visualize functions, compare curves, and inspect intercepts or intersections before moving to algebraic detail.
How it works
Enter one or more expressions, set a useful x/y window, and let the page sample points for plotting. Numerical intercepts and intersections depend on the visible range, so adjust the window before interpreting missing or extra roots.
When to use
Use it for classroom sketches, quick sanity checks, comparing transformations, or finding approximate crossing points before solving an equation exactly.
Common mistakes to avoid
- Assuming a root is absent when it is outside the current viewing window.
- Using too wide a range for a function with sharp behavior near an asymptote.
- Comparing curves without checking scale, units, and axis labels.
- Treating sampled graph points as exact symbolic solutions.
Interpretation and worked example
Start with a narrow window around the expected behavior, then zoom out if the curve is clipped. If two graphs appear to touch, use the intersection list as an approximation and verify exact values with an algebra tool when needed.
See also
FAQ
How does the calculator find intersections or x-intercepts?
The viewport is sampled at fixed intervals to detect sign changes. Each bracket is refined with up to 40 bisection iterations, and the step-by-step log lists the interval width and g(x) values so you can follow the convergence.
What happens when I switch between degrees and radians?
Trigonometric expressions are converted internally according to the selected unit. Choosing degrees makes sin(90) evaluate to 1, while radians expects values such as sin(pi/2), so the graph remains correctly scaled.
What should I enter first for graphing?
Enter one expression such as sin(x), x^2 - 1, or x^3 - x with a sensible x-range first. Add more functions only after the first curve, axes, and detected points are readable.
Why can graphing results differ from nearby tools?
Differences usually come from expression syntax, degree/radian mode, graph window, and sampling resolution. Match those assumptions before comparing roots, extrema, or intersections.
How should I judge the detected points?
Use the table and step log together. Points are found by sampling the viewport and refining brackets, so a tighter window or higher sample count can reveal roots or intersections that a broad view misses.