Graphing calculator

How to use the graphing calculator

• Type a function of x as you would on a calculator: x^2 - 4x + 3, sin(x), 1/x, sqrt(x), e^x. Implied multiplication works, so 2x and 3(x + 1) are fine.
• Up to five functions are drawn on the same axes, each in its own colour.
• Drag to move the graph, and scroll or pinch to zoom. On the keyboard, the arrow keys move,
  • and − zoom and 0 resets the view.
• Draw the derivative with the box beside the function. It is drawn dashed in the same colour, and its formula is written out beside it.
• The address bar holds the functions and the view, so a link shows the same graph again.

The points under the graph

Below the graph are the points you can read off in the part of the graph that is in view:

• Zeros, where the curve crosses the x-axis.
• Minimum and maximum points, where the curve turns.
• The y-intercept, where x = 0.
• Intersections of two functions, where they have the same value.

The points are found numerically and rounded to four decimals. A point outside the view is not listed, so move or zoom the graph to look elsewhere. A solution such as x = √2 is therefore shown as 1.4142; the equation solver gives the exact value.

Worked example

The page opens with f(x) = x² − 4x + 3 and g(x) = x − 1. The list shows:

• f has zeros at x = 1 and x = 3, and a minimum at (2, −1), halfway between them, which is where a quadratic with two zeros has its turning point.
• f and g meet at (1, 0) and (4, 3). Those are the solutions of x² − 4x + 3 = x − 1, read off rather than worked out.

Tick the derivative of f and f′(x) = 2x − 4 is drawn: a straight line crossing the x-axis at x = 2, exactly where f has its minimum. That is the link between a function and its derivative, visible in one picture.

Parameters and sliders

Type another letter in a function, like a and b in a*x^2 + b*x, and each letter gets a slider below the functions. Drag it and watch the curve change shape as you drag: what a does to the width of the parabola, what b does to where it sits. The same letter in two functions is the same slider, so both curves move together.

Each slider starts at 1 and runs from −10 to 10 in steps of 0.1. The value can also be typed in the field beside it, and the range is set in the three small fields below. The values are in the link, so a shared link shows the curve with the same a and b.

Tangent

Pick a function under Tangent and type an x, such as 2, 0.5 or pi/2, or click the curve where you want it. The tangent is drawn dotted in the function’s colour, with a mark at the point, and its equation is written below. The slope comes from the exact derivative rather than an estimate from the picture, so the tangent to x² at x = 2 is exactly y = 4x − 4. Where the function has no derivative, as 1/x at x = 0, the page says there is no tangent.

What it does not draw

Only x runs along the axis; any other letter gets a slider instead. Curves that are not functions of x, such as the circle x² + y² = 1, are not drawn, and trigonometry works in radians. Where a function is undefined, as 1/x at x = 0 or tan(x) at its vertical asymptotes, the curve breaks instead of drawing a line straight across.