# Graphing calculator

> Plot several functions, polar curves and parametric curves; pan and zoom; find roots, intersections, turning points and areas.

Interactive version: https://www.calcopenly.com/math/graphing-calculator
Subject: Math calculators

Plot up to six functions of x on one set of axes, alongside polar curves such as r = 1 + cos(θ) and parametric curves such as (cos(t), sin(2t)). Drag to pan, scroll or pinch to zoom, and hover to read each function's value, which is recomputed in 50-digit decimal at the pointer.

Below the plot, the calculator lists the roots, turning points and intersections visible in the current window, found by bisection on sign changes. It can also plot the derivative of the first function and shade the area under it between two limits, reporting the integral by Simpson's rule.

Trigonometric functions take radians here, as in most graphing software; type pi/2 rather than 90.

## Inputs

- **f(x)**
- **x**

## Results

- f(x) — main result

## Worked examples

### Cubic at 2

- f(x): x^3 - 3x
- x: 2
- **f(x): 2**
- Checked against: 8 − 6

### Sine at π/2

- f(x): sin(x)
- x: pi/2
- **f(x): 1**
- Checked against: sin(π/2) = 1; the plotter works in radians

### Rational function

- f(x): 1/(x-1)
- x: 3
- **f(x): 0.5**
- Checked against: 1/2

## Questions

### How do I find where two graphs intersect?

Enter both functions and look at the 'Points of interest' table under the plot. It reports numerical candidates in the visible window, with coordinates rounded to 8 significant figures; this display precision does not guarantee that accuracy. Sampling can miss tangencies or closely spaced intersections. Zoom or pan to inspect the region, and check important solutions in the original equations.

### Can it plot polar and parametric equations?

Yes. Start a line with r = to plot a polar curve in θ (for example r = 2 sin(3θ) draws a three-petal rose), or write a pair in brackets such as (cos(t), sin(t)) for a parametric curve, traced for t from −2π to 2π.

### Why does tan(x) have gaps?

tan(x) has vertical asymptotes at x = π/2 + kπ, where it jumps from +∞ to −∞. The plotter breaks the line wherever consecutive samples jump by more than one and a half screen heights, so it doesn't draw false vertical lines across the asymptotes.

### How accurate is the graphing calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 3 worked examples whose answers come from independent sources; for example, “Cubic at 2” is checked against 8 − 6.

### Where does the method come from?

OpenStax Precalculus 2e — Functions, function notation and graphs; Implementation: adaptive sampling with discontinuity breaks; values at a point recomputed in 50-digit Decimal.

## Sources

- [OpenStax Precalculus 2e — Functions, function notation and graphs](https://openstax.org/books/precalculus-2e/pages/1-1-functions-and-function-notation)
- Implementation: adaptive sampling with discontinuity breaks; values at a point recomputed in 50-digit Decimal