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
How it's calculated
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 do first on this page?
Start with the minimum required inputs or the first action shown near the primary button. Keep optional settings at defaults for a baseline run, then change one setting at a time so you can explain what caused each output change.
Why does this page differ from another tool?
Different pages often use different defaults, units, rounding rules, or assumptions. Align those settings before comparing outputs. If differences remain, compare each intermediate step rather than only the final number.
How reliable are the displayed values?
Values are computed in the browser and rounded for display. They are good for planning and educational checks, but for regulated or high-stakes decisions you should validate assumptions with official guidance or professional review.