Graphing Calculator – Interactive Function Plotter

What is a graphing calculator?

A graphing calculator is a tool that plots functions of a variable – most often functions \( y = f(x) \) – on a coordinate plane. Instead of computing individual values by hand, you see the entire curve and how it behaves across an interval.

Graphs help you detect zeros, maxima/minima, asymptotes, oscillations, and trends at a glance. They are essential in algebra, trigonometry, calculus, probability, and many applied fields.

Supported function syntax

This graphing calculator expects functions of a single real variable x. You can combine:

• Basic operations: +, -, *, /, ^ for powers.
• Parentheses: ( ) to control precedence.
• Constants: pi (π), e (Euler’s number).
• Functions (case-insensitive): sin, cos, tan, log (natural logarithm), sqrt, abs, exp.

Examples of valid inputs:

• x^2 - 4*x + 3
• sin(x) / x
• exp(-x^2)
• sqrt(abs(x))
• 2*cos(3*x) + 0.5

Viewing window and scaling

The viewing window determines what portion of the plane you see. If a function looks “flat” or you cannot see key features, it is almost always a window issue.

• x-min / x-max control the horizontal range on the x-axis. For example, \([-10, 10]\) is a common default.
• y-min / y-max control the vertical range on the y-axis. Adjust these to zoom in around interesting parts of the curve.
• The zoom buttons rescale the window symmetrically around the origin, halving or doubling both ranges.
• On desktop, you can click and drag the canvas to pan the window in x and y (within reasonable limits).

Sampling and numerical limitations

The calculator draws curves by sampling the function at many points between x-min and x-max. For each visible x, it evaluates \( f(x) \), converts it to a canvas coordinate, and connects consecutive valid points.

This is fast and works well for smooth functions, but there are limitations:

• Discontinuities (for example at vertical asymptotes) can cause jumps. The tool detects large vertical jumps and avoids drawing lines across them, but a very coarse step might miss fine details.
• Very steep or oscillatory functions may need a narrower x-range or finer sampling to reveal their structure clearly.
• Values that overflow or underflow numerically are ignored to keep the graph readable.

When in doubt, combine visual inspection with analytical work: compute limits, derivatives, or exact intercepts using algebra or a CAS when precision matters.

Worked examples

1. Quadratic vs. cubic

Try plotting f₁(x) = x^2 and f₂(x) = x^3 on the same window \([-4, 4]\) by \([-4, 4]\). You will see the parabola opening upwards and the cubic changing concavity at the origin.

2. Sine wave and damping

Enter f₁(x) = sin(x) and f₂(x) = exp(-0.2*x)*sin(x) on \([-10, 10]\) by \([-2, 2]\). The second curve shows a damped sine wave, demonstrating how an exponential envelope affects an oscillatory signal.

3. A function with a vertical asymptote

Consider f₁(x) = 1/(x-1). On \([-4, 4]\) by \([-10, 10]\), the graph shows a vertical asymptote at \(x = 1\) and two separate branches. The plotting routine skips the infinite jump at the asymptote to avoid drawing a meaningless vertical line.

FAQ – using the graphing calculator