Online graphing calculator, free and with nothing to install

This online graphing calculator plots the curve of a function on a set of axes, the way a TI-84 Plus CE or a TI-Nspire CX would.

You type f(x), the curve appears as you type, and you can then pan it, zoom it, trace it point by point and look for its zeros, its extrema and its intersections.

Six curves can share the same axes, each in its own color, which is plenty to compare a function with its tangent line, a parabola with a line, or a family of curves that differ only by one parameter.

A formula may contain letters: write ax+b and two sliders appear under the function list. Dragging them rotates and shifts the line before your eyes (one button even sweeps them on its own). And an f′ button on each row adds the curve of the derivative, dashed.

The graph can be saved as an image, with a legend for the curves, ready to paste into homework; and a link reopens it exactly as it stands on screen, so you can send it to someone.

Nothing to install, nothing to sign up for: the page works on a computer, a tablet and a phone, with its own keypad for the keys a system keyboard does not have (x, π, √, the exponents).

For numeric work alone (trigonometry, logarithms, statistics, no plotting) the site’s online scientific calculator is made for that, and both pages use exactly the same way of writing expressions.

Help for the graphing calculator

Plotting a first curve

Type the expression of the function in the f₁(x) = field at the top of the calculator. The curve is drawn as you type: there is no key to press.

The + Add a function button opens a second row, then a third, up to six. Each row has its own color, shown by the dot in front of it; clicking that dot hides the curve without erasing the formula, which is handy for comparing two curves by showing them in turn.

The cross at the end of the row deletes the function. The Clear all button starts again from an empty calculator.

If the expression contains a mistake, a message explains it under the field, but only after a short pause, to let you finish typing: sin(x) goes through s, si and sin, none of which means anything yet.

Writing a function

The variable is always written x. The four operations are written +, -, × (or *) and ÷ (or /), and the order of operations is respected: 2+3x means 2 + (3×x).

Implied multiplication

The multiplication sign may be left out before the variable, a parenthesis, a constant or a function: 2x, 3(x+1), 2π and 2xsin(x) are all understood. That is how mathematics is normally written, and it is what the calculator expects.

Powers

Powers use ^: x^3. The x² and x³ keys write the exponents ² and ³ directly, which read better: x²-2 is clearer than x^2-2. The exponent may be negative (x^-2) and powers associate to the right, so 2^3^2 is 2⁹.

Roots

The square root is written √( or sqrt( or squareroot(, the cube root ∛( or cbrt(, and the nth root root(x,n).

Unclosed parentheses

Parentheses left open are closed for you: sin(x plots the same curve as sin(x). You never need to count them.

The period and the comma

The decimal separator is the period: write 1.5x. The comma separates the arguments of a function, as in log(x,2) where it separates the number from its base.

Sweeping a parameter

A formula does not have to contain only numbers. Write ax+b instead of 2x+3, and two sliders appear under the function list: dragging them rotates and shifts the line live. It is the shortest way to understand what the coefficients of a linear function do.

The available letters

Twelve letters can be given sliders: a, b, c, d, h, k, m, n, p, q, r and t. They are the ones textbooks use (the coefficients a, b, c, the slope m and the intercept b, the shift h and the height k, the index n).

Other letters are refused, except x for the variable and e for Euler’s number.

There is nothing to declare: the slider for a letter appears as soon as a formula uses it, and disappears as soon as you delete it. Two functions using the same letter share the same slider.

Setting, bounding, animating

The field to the left of the slider gives the exact value, and accepts an expression: write π/2 rather than its decimal value. A value outside the interval pushes the bound out instead of being forced back in.

The two small fields on the right set the interval swept, −10 to 10 by default. Narrow it for fine tuning, widen it to explore.

The ▶ button starts the animation: the parameter travels back and forth between its bounds in ten seconds, and the curve deforms on its own. That is where you see what a parameter really does (a parabola opening up and flipping over when a changes sign, a sine wave sliding along the axis. A second press stops it).

A few formulas worth trying

ax+b (the linear function): a tilts the line, b moves it up and down.

a(x-h)²+k (vertex form of a quadratic): h and k are exactly the coordinates of the vertex, and a is how wide the parabola opens.

a·sin(bx+c) (amplitude, frequency and phase shift of a sine wave): one slider each.

x^n (animate n between 1 and 5 to watch the curve go from a line to a parabola, then to a cubic).

Panning and zooming

The axes are handled directly, without going through any settings:

With the mouse: click and drag to pan; turn the wheel to zoom around the pointer. Hovering the graph shows the pointer’s coordinates in the top right.

With a finger: drag to pan, and pinch to zoom.

With the keyboard: click the graph first to give it focus, then use the arrow keys ← → ↑ ↓ to pan, and + and − to zoom.

The + and − buttons on the graph zoom around the center, which is sometimes handier than the wheel when you want to keep the origin where it is.

Setting the window

Four buttons in the top bar cover the most frequent cases:

Standard goes back to the −10 to 10 window on both axes, the one you almost always start from.

Square sets the y range so that one unit measures the same number of pixels on both axes. That is what it takes for a circle to look round, and for two perpendicular lines to look perpendicular: without it, the line y = x does not make a 45° angle.

Trig sets the window from −2π to 2π and graduates the x-axis in multiples of π. An axis marked 3.14 / 6.28 says nothing; the same axis marked π / 2π reads itself.

Zoom Fit lines the y range up with the values the curves actually take in the window. It is the button to use when the curve is flat or runs off the screen and you do not know what scale to choose.

The Window tab, under the graph, gives access to the four bounds Xmin, Xmax, Ymin and Ymax. They accept an expression: writing -2π is shorter, and more exact, than copying its decimal value.

Tracing the curve

The Trace button puts a point on the curve and shows its coordinates. The point follows the mouse or the finger; with the keyboard, the ← and → keys move it one pixel, and with Shift ten pixels at a time.

When several curves are plotted, the ↑ and ↓ keys move from one curve to another, just like on a physical graphing calculator.

When the point reaches the edge of the window, the window itself slides: you can follow a curve as far as you like without losing sight of it. The Keep this point button copies its coordinates into the results history, below the calculator.

Zeros, extrema and intersections

The Key points tab searches on its own, within the window shown:

The zeros of the function, that is the x-values where the curve crosses the horizontal axis. They are the solutions of the equation f(x) = 0.

The local extrema, that is the points where the curve turns around: it was rising and starts falling (a maximum), or it was falling and starts rising (a minimum). The graphing calculator gives their coordinates. A place where the curve merely flattens without turning around — like x³ at 0, which levels off for an instant before carrying on up — is not an extremum, and is therefore not reported.

The y-intercept, that is f(0), when x = 0 is inside the window.

The intersections with a second curve, chosen from the list. This is the graphical way of solving an equation f(x) = g(x): plot both sides and read off the x-values of the common points.

The Center button brings the point found to the middle of the window, so you can zoom in on it. The Keep button files it in the results history.

One important point: the search only covers the visible part of the curve. A function like the sine has infinitely many zeros; widening the window brings out more of them.

Derivative, tangent line and the curve of f′

The curve of the derivative

The f′ button, to the right of each function row, adds the curve of its derivative to the graph (dashed, in the same color as the function, and underneath it so as not to hide it).

It is the whole increasing-decreasing analysis in a single picture: where the dashed curve is above the x-axis, the function rises; where it is below, the function falls; and where it crosses the axis, the function has an extremum. Try it with x³-3x: the derivative is the parabola 3x²-3, and its two zeros at −1 and 1 land exactly on the maximum and the minimum of the cubic.

The two curves are controlled separately: the color dot hides the function without touching its derivative, and the f′ button removes the derivative without touching the function.

The derivative and the tangent line at a point

The Derivative and integral tab asks for an x-value and gives, for the curve being studied, the value f(x), the derivative f′(x) and the equation of the tangent line at that point.

The Draw the tangent line button puts that line on the graph, dashed. It is the quickest way to check a tangent line equation worked out by hand: if the line you computed hugs the curve at the right point, the work is correct.

The derivative is obtained numerically, by a five-point centered difference, both for the curve and for the value at a point. It is exact up to rounding error, which is more than enough to check homework, but it is not computer algebra: the calculator plots the derivative, it does not give its formula.

Integral and area under the curve

Two bounds a and b are enough to get the integral of the function between them, computed by Simpson’s rule. The Shade the area button paints the matching surface between the curve and the x-axis.

That is what makes visible a property the calculation alone leaves abstract: the area below the axis counts as negative. Take sin(x) in radians, integrate from 0 to π and you get 2; from 0 to 2π you get 0, because the two humps cancel out.

The table of values

The Table of values tab gives the values of the plotted functions for a run of evenly spaced x-values, set by a start, a step and a number of rows. It is the equivalent of the TABLE key on a graphing calculator.

It serves two purposes: building a table of values asked for in an exercise, and spotting a sign change (and therefore a zero) between two consecutive rows. The Send to the history button copies the whole table below the calculator, from where it can be exported as CSV for a spreadsheet, or as PDF.

Saving the graph, or sharing it

The image

The Save image button, under the graph, produces a PNG file ready to be dropped into homework, a lab report or a lesson. The image is saved at twice the size shown: it stays sharp once printed.

It carries a legend in the top left (each curve’s equation, in its color, and the value of the parameters at the time of saving). A curve without its equation means nothing once it leaves the page; the legend saves you from writing it out beside. In the bottom right is the address of this page: a graph travelling through homework or a slide deck therefore says where it came from, and where to make another.

The zoom buttons and the Trace button do not appear on it, unlike what a screenshot would have carried along. And the image always comes out on a light background: a dark graph printed on white paper is unreadable.

The link

The Share link button builds an address containing the whole graph: the formulas, the window, the angle mode, the parameters and their values, the derivative curves shown. Whoever opens it gets exactly the same screen, and can carry on zooming, panning and studying from there.

The link is copied to the clipboard and replaces the page address, so you can also grab it from the browser bar. It is shown in a field as well, with a Copy button to pick it up at any time (useful when the browser refuses access to the clipboard, which happens in private browsing on some phones: the link is then selected, and copying it from the keyboard is all that is left to do).

This is what a teacher sends to a class to set an exercise on a specific graph, and what a student sends back to show where they got to. Nothing is stored on the site: everything is in the address, and it works indefinitely.

Opening a link you received does not touch your own graph: if you had switched “Keep my work” on, this device’s memory is simply paused for that visit, and you get your curves back the next time you load the page without the link.

Keeping your work from one visit to the next

The calculator opens on the same parabola every visit. The “Keep my work” switch, at the very bottom of the calculator, changes that: your functions are then saved in your browser, along with the window, the angle mode, the grid, the graduations, your parameters and their sliders, as well as the fields of the study panel (the point examined, the integral bounds, the table settings). You get all of it back exactly as it was on your next visit, even after closing the tab.

Nothing is sent over the internet: everything stays in this browser, on this device (so your work does not follow you from one computer to another — for that, send yourself the share link).

The switch is off to begin with, because on a shared machine — a school library or a classroom — finding the previous visitor’s curves would not be pleasant. Switching it off erases at once whatever had been kept. And if you clear everything before leaving, an empty screen is what you get back: the memory keeps what you leave behind, not what the page started with.

Degrees or radians

The angle mode completely changes the look of a trigonometric curve. In radians, sin(x) completes one period over 6.28 units; in degrees it needs 360, and on a −10 to 10 window the curve looks almost straight.

The mode is shown in the top bar and is radian by default, as on every graphing calculator and as in precalculus and calculus courses. The ° suffix forces degrees for a single number, even in radian mode: sin(30°) is 0.5.

Every accepted notation

Spelled-out names lead to the same functions: squareroot(, cuberoot(, absolute(, modulus(, remainder(, exponential(, arcsin(, int( and ceil( are understood.

The computer keyboard

The function fields can be filled straight from the computer keyboard, without going through the calculator keypad. A few useful equivalents:

* ⇝ ×

/ ⇝ ÷

^ ⇝ power

Enter ⇝ open a new function row

On the graph, once it has focus: ← → ↑ ↓ to pan, + and − to zoom.

The calculator keypad, folded away by default on a computer, opens with the Show the keypad button. It is open from the start on a phone: the keys x, π, √ and the exponents exist on no system keyboard.

Examples, from algebra to calculus

Pre-algebra and Algebra 1

Plot 2x+1: the line crosses the y-axis at 1 and rises by 2 whenever x increases by 1. Add -x+4 and ask for the intersections: the common point (1, 3) is the solution of the system. That is the graphical solution of a system of two equations.

Plot x² and then x³ on square axes, and compare their shapes either side of the origin.

Algebra 2 and Precalculus

Plot x²-4x+3. The key points tab gives the zeros 1 and 3, and the minimum at the vertex (2, −1). You get the vertex form (x-2)²-1 without having completed the square.

Plot 1÷x: the curve is cut in two near 0, exactly where the function is undefined. No vertical line joins the two branches (that is an asymptote, not a value).

Check a tangent line equation: enter the function, choose the point, read the equation given and draw the tangent line.

Write a(x-h)²+k and animate a: the parabola opens, narrows, then flips over when the coefficient goes negative. Then set h and k: they are the coordinates of the vertex, and it shows.

Calculus

Plot exp(-x²), the bell curve, and integrate it from −2 to 2: you get 1.7641628. Shade the area to see what that means.

Plot ln(x) and x-1: the two curves touch at (1, 0) without crossing, which is the graphical translation of the inequality ln(x) ≤ x-1.

Show the derivative of x³-3x²+2 with the f′ button: the increasing and decreasing intervals read themselves, the function rising wherever the dashed curve is above the axis.

Switch to Trig mode and plot sin(x) and cos(x): the intersections land at π/4 and its multiples.

College

Compare x, x², exp(x) and ln(x) on one window, fitting the y range: the relative growth rates show at a glance.

Plot sin(x)÷x: the curve passes through 1 at zero, although the function is undefined there (the limit reads itself off the graph).

Plot tanh(x) and its asymptotes 1 and -1.

The results history

Every value kept with a Keep button (a zero, an extremum, an intersection, a derivative, an integral, a traced point, a whole table) is recorded in the “Results history” table below the calculator, together with the function concerned. Every cell there can be edited, and a notes column lets you annotate rows.

You can export that table as CSV to open it in a spreadsheet, or generate a PDF document.

What this calculator does not do

It plots functions of the form y = f(x), and nothing else: no parametric curves, no polar coordinates, no sequences, no two-variable statistics, no implicit equations such as x²+y²=1. A circle can therefore only be obtained by plotting its two half-circles, √(1-x²) and -√(1-x²), on square axes.

Nor does it do computer algebra: it plots the curve of the derivative but does not give its formula, does not factor, and does not solve an equation symbolically. The values it shows are obtained numerically, with the precision of floating-point numbers (plenty for an exercise, never for a proof).

To solve exactly, see the quadratic equation solver, the linear equation solver or the slope calculator.

Who it is for

High school students, first of all: reading increase and decrease off a graph, checking a tangent line, solving an equation graphically and building a table of values run right through Algebra 1, Algebra 2, Precalculus and Calculus — and a graphing calculator is not always within reach.

College students, to picture a function before studying it (a curve says in one second what a sign chart takes a page to establish).

And teachers, who can project the axes and bring out one curve after another, or show live what a parameter does to the shape of a family of curves.