Graphing Calculator

A graph is a check, not a picture

Most of the time a student plots something to find out whether they were right. The algebra said the curve turns at two — so does it? The formula library holds the expression but not the shape, and the shape is the part that is hard to picture from symbols alone. The graphing calculator takes an equation and draws it, which sounds like a small thing until you are trying to see whether a negative sign belongs inside or outside a bracket.

It opens from four places, and which one you use depends on where the equation came from. A blank graph is the tile on the Home screen. An equation you were already reading can be sent straight from the formula library, keeping its symbols and the values bound to them. A link opens a specific graph, which is what the share card carries, so a graph sent to a classmate arrives as a graph rather than a photograph of one. And pasting an equation into an empty graph works, because that is how most equations reach a phone in the first place.

Three kinds of thing, and how you tell them apart

There is no mode to select. What you type decides what is drawn.

• An equation in x and y draws a curve on the plane.
• An equation beginning z = is read as a surface, and it takes the whole canvas, because a surface needs the room.
• Something inside square brackets, like [1 2; 3 4], is read as a matrix — and a matrix can be looked at two ways: as a heat map, where each cell is shaded by its value, or as a surface, where the same numbers become heights.

The syntax is ordinary mathematical notation rather than a programming language. 2x means two times x, and x^2 is a square. The on-screen keypad inserts the unambiguous form of each symbol, because a few of the prettier characters cannot be parsed as themselves — the square-root sign is one, and it inserts sqrt( instead, so that what you see on screen and what the parser reads do not drift apart.

Why the surface view fits the way it does

Drawing a surface on a phone means choosing a camera, and the choice is less obvious than it looks. The fit is computed from a bounding sphere around the plotted points rather than a box, because no rotation can move a point off a sphere. A box-sized fit lets the shape slide out of frame as you turn it, and it drew a paraboloid as a narrow spike. The box and the three axes are drawn underneath the wireframe, and they are taken from the fitted points rather than the raw ranges, so they stay correct at every angle without anything needing to know the scale.

The opening angle is not the mathematically tidy one. A surface was rendered at several pitches and looked at, because from a low angle a paraboloid's four corners read as a crown and from a high one a wave flattens into a plane. What is stored is the angle that was checked by eye rather than the one that was easiest to derive.

What it does not do yet

The surface view is a wireframe with no hidden-surface removal, so a closed shape is see-through. A shallow block reads well; a tall one becomes a tangle, because the top, the base and all four walls are drawn over each other. A matrix is also a single height per point, so it can make a raised block but never a closed cube.

Polar, parametric and contour plots, vector fields and symbolic algebra are designed but not built — they are later slices of a nine-part plan, and only the first has shipped. A small number of the formula library's plot-ready expressions use notation the parser does not yet read, such as an infinity or a derivative prime. Those are reported as what they are rather than drawn wrongly.