Graphing calculator

What you can type

• Functions: y = x^2 - 3x, sin2x, e^-x, |x - 1|, sqrt(x), ln x, log_2(x)
• Your own functions: f(x) = x^3 - 3x, then f(x - 2) + 3 or f'(x)
• Sliders: any letter you have not defined, like a in y = a(x - h)^2 + k, offers a slider; ▶ animates it
• Points and lists: (2, 3), n = [1...10], (n, 2n + 1). Drag a point like (h, k) to move its sliders; a label such as (${h}, ${k}) shows its values
• Piecewise and restrictions: {x < 0: -x, x^2}, y = sqrt(x) {0 <= x <= 4}
• Calculator: int_0^2 x^2 dx, sum_(n=1)^(10) n^2, mean(y_1), nCr(5, 2)
• Started in Desmos? Import from Desmos brings in copied expressions, sliders and table cells, and tells you anything it couldn't bring in.

Modelling and regression

Paste two columns from a spreadsheet into an expression line to make a table with lists x_1 and y_1, then type a model with ~:

• y_1 ~ m x_1 + b: the regression line of y on x, with r (the product moment correlation coefficient) and R²
• y_1 ~ a e^(k x_1): exponential growth and decay; y_1 ~ a x_1^n: a power model
• y_1 ~ a sin(b x_1 + c) + d for data that repeat, such as tide heights or hours of daylight

A Level asks you to use logarithmic graphs to estimate the parameters of models like y = a x^n and y = k b^x. Switch on log scales in More → Settings: when the model is right, the points lie close to a straight line.

Under every fit, What this model means explains each parameter in your context (from the axis labels in Settings), how to read r and R², whether the residuals show a pattern, and where the model can be trusted. Compare models under a table fits the standard models to the same data, and residual plot draws the residuals in a small second graph. + Add → Image puts a photo behind the axes, so you can read points off a curve and fit a model to them.

Practise it: exponentials and logarithms · IAL S1 correlation and regression · correlation and regression calculator

Reading the graph

• Tap a curve to select it: grey dots mark its roots, turning points, y-intercept and intersections with other curves. Tap a dot for its coordinates, or drag along the curve to trace it.
• Drag to move; pinch, scroll or use + and − to zoom; ⌂ resets, ⤢ fits your data, □ makes the axes square.
• Keyboard: focus the graph, then arrow keys move and + / − zoom. More → Data table lists the values and points of interest.
• Undo and Redo (the arrows by More, or Ctrl+Z and Ctrl+Shift+Z; ⌘ on a Mac) take back changes, even loading an example over your graph.
• In More, Share link puts the whole graph in the link (nothing is uploaded, so pictures you add are not included); Export downloads a PNG or an SVG, with your residual plots if you want them. Your graph, with its pictures, is also kept in this browser.
• On a phone the graph stays at the top while you scroll your lines, and the maths keypad (More → Maths keypad) gives you ^, √, π, θ, x₁ and ~ without switching keyboards. Select a curve and tap table of values for a table.
• Save keeps it in My graphs with a free account (up to 200 graphs, and up to 5 pictures per graph), to open, rename or delete on any device.
• Pictures are made smaller in your browser before they are used (at most 1600 pixels across), which also removes the photo's hidden details such as its location.

Practising for a calculator paper? In the exam you use your own calculator, and it must follow your board's rules (for UK A Level, the JCQ rules). See the A Level calculator guide and which calculators are allowed. This grapher is our own tool; the ~ syntax follows the convention many graphing tools use.

Practise with it

• Graphs and transformations: drag the sliders in y = a f(x - h) + k
• Trigonometry: sketch y = 3sin(2x) + 1 in degrees or radians (More → Angles in degrees)
• Exponentials and logarithms: y = e^x, y = ln x and their asymptotes
• Differentiation: graph f(x) and f'(x) together and check your stationary points
• Numerical methods: see where a root lies before you use a change of sign or Newton–Raphson
• Correlation and regression: fit the line of y on x and read off r

Sketch it first, then check it here: in the exam you sketch by hand.