Grapher syntax reference

The grapher reads the same plain notation you would type into a calculator. This page is the full reference: the kinds of plot, how expressions are parsed, and every function and constant. The same syntax is used inside notes in graph blocks.

Kinds of plot#

The grapher decides what to draw from the shape of what you type.

Each row holds one expression with at most one =, <, >, <= or >=. A double inequality such as 1 < x < 3 needs two rows.

Functions of x#

Write the right-hand side on its own or with y = in front. Both of these plot the same curve.

x^3 - 3x + 1
y = x^3 - 3x + 1

When you paste LaTeX, a leading f(x) = (or g(x) =, h(x) =) is turned into y =. Typed directly, write y = instead. Functions of $y$ are plotted sideways with x = …, or by typing an expression in y alone:

Implicit equations#

Any equation involving both x and y is drawn as an implicit curve, so circles, ellipses and curves that are not functions all work.

An expression with both x and y but no = is ambiguous, so the grapher asks you to write it as an equation, such as x^2 + y^2 = 4.

Inequalities#

Use <, >, <= or >= (or ≤ and ≥). The region where the inequality holds is shaded. Strict inequalities draw a dashed boundary; <= and >= draw it solid.

Inequalities in one variable, such as x > 1, shade a half-plane.

Polar curves#

Use θ for the angle, or type theta instead. The curve is drawn for $0 \le \theta \le 2\pi$.

An expression in θ alone, without r =, is also drawn as a polar curve.

Parametric curves#

Write a pair (x(t), y(t)) that uses t. The curve is drawn for $0 \le t \le 2\pi$.

To cover a different range, scale t inside the expressions: (2t, 4t^2) covers $0 \le x \le 4\pi$.

Points#

A pair of numbers is a point, drawn as a dot labelled with its coordinates. Coordinates can be expressions such as (pi/2, 1), and they can use sliders, so (a, a^2) moves along a parabola as you drag a. Points cannot depend on x or y.

Sliders#

A row of the form letter = number defines a slider. The value can be any constant expression, such as k = pi/4; once you drag the slider, the row shows the number. Any other letter used in an expression gets a slider automatically, starting at 1.

a = 2
y = a sin(x - b)

Slider names are single letters other than the reserved letters below, upper or lower case, or a letter followed by digits such as a1 or k2. Because a letter followed by digits is always read as one name, write 2x rather than x2.

Reserved letters#

Everything else is free to use as a slider. Function names and the constants pi and tau are read as themselves, not as sliders.

How expressions are read#

The parser follows the conventions you use on paper.

Function names and constants are recognised even when run together with other letters, so sinx is $\sin x$ and 2pix is $2\pi x$. This stops working once a digit is in the run: sin2x is read as a single slider name, so write sin 2x. When in doubt, add a space or brackets.

Numbers can be written as 3, 0.25, .5 or in standard form, 1.5e3. Unicode −, ≤, ≥, ·, ×, π, τ and θ are all accepted.

Constants#

Functions#

All trigonometric functions use radians.

For logarithms in another base, use the change of base: $\log_3 x$ is ln x / ln 3.

Examples#

Graph blocks in notes#

Notes draw figures with a ```graph fenced block: one instruction per line. Any grapher expression can be a line, plus a few instructions for framing the figure. Blank lines are ignored and lines starting with # are comments.

Numbers in view, fill and coordinates can be expressions such as pi, pi/2 or 2pi. In view and fill each number must be written without spaces, so use pi/2 rather than pi / 2. Segments and arrows need a space either side of -- or ->. A fill only works with a curve of the form $y = f(x)$.

Every line that draws something, or sets a constant, is listed under the figure as its caption, so readers can see exactly what is drawn. Figures in notes do not show points of interest; choose Open in grapher to explore them.

```graph
view 0 2pi -1.5 1.5
rad
y = sin x
y = cos x
fill 0 pi/2 y = sin x
(pi/4, 0) -- (pi/4, 0.7071)
```

That block draws this:

The window is padded slightly and widened to fit the figure's shape, so curves never touch the frame. Lines that cannot be plotted are reported by pnpm notes:check and shown under the figure, so a broken graph is never silently empty. See Note format for when to use a graph and when to draw a diagram.