Graphing Calculator

Type an equation such as y = x² − 2x − 3 into a line and the graph is drawn as you type. Add more lines to draw several graphs at once; the answer box gives the x-intercepts, the vertex or turning points, and where the graphs cross, with the working.

Graphing Calculator

Drag to move · scroll or pinch to zoom · tap a curve for its point

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How to type an equation

Write it the way you would on paper. Side by side means multiply, so 2x, 3(x + 1) and 2sin(x) all work, and you can leave the brackets off a simple function: sin 2x is sin(2x). Use ^ for a power (x^3) or the x² key. Each line is drawn in its own colour; press Enter to start a new line, click the coloured dot to hide or show a graph, and × to delete it.

Type	What you get
y = 2x + 1	A straight line, with its slope and both intercepts
y = x^2 − 2x − 3, or just x^2 − 2x − 3	A parabola, with the vertex and the roots by the quadratic formula
x = 3	A vertical line (x = something in y draws a sideways curve)
x^2 + y^2 = 25	A circle; any equation using both x and y is drawn where both sides are equal
y < 2x + 1, x^2 + y^2 ≤ 16	A shaded region; type <= and >= for ≤ and ≥
(2, 3) or (1, 1), (4, 5)	Points, labelled; two points also give the distance, midpoint and slope
f(x) = x^3 − x, then y = f(x − 1)	A function you name once and use again
a = 2, then y = a sin(x)	A slider for a; drag it to see the graph change
y = sqrt(x) {x < 4}	Only the part of the graph where the condition in { } is true
r = 2sin(4θ)	A polar graph, for θ from 0 to 12π
(cos t, sin t)	A parametric curve, for t from 0 to 2π; add {0 ≤ t ≤ 10} for another range
Add table	A table of x₁ and y₁ values, plotted as points; paste two columns from a spreadsheet to fill it
y₁ ~ mx₁ + b (or y_1 ~ m x_1 + b)	The best-fit line through the table, with R²; any formula works, such as y₁ ~ a·b^x₁
sqrt(−4), (1 + 2i)(3 − i)	With Complex mode on (in the settings): complex answers, here 2i and 5 + 5i, plotted as points
2^10 / 4	A plain sum, worked out on the line: = 256

The functions it knows are sin, cos, tan, sec, csc, cot and their inverses (asin or arcsin and so on), sinh, cosh, tanh, sqrt, cbrt, ln, log (base 10, or log(8, 2) for base 2), exp, abs or |x|, floor, ceil, round, sign, mod(a, b), min, max, nthroot(x, n) and the factorial x!. The constants are π (or pi) and e.

Reading the answer

The answer box describes the line you're in. For a straight line or a parabola it reads the equation off exactly and solves it with these formulas, showing each step:

straight line: y = mx + b crosses the x-axis at x = −b ÷ m parabola: y = ax² + bx + c vertex at x = −b ÷ 2a roots: x = (−b ± √(b² − 4ac)) ÷ 2a circle: (x − h)² + (y − k)² = r² centre (h, k), radius r

The number under the square root, b² − 4ac, tells you how many times a parabola meets the x-axis before you draw it: twice if it's positive, once (just touching) if it's 0, and never if it's negative. For other graphs, such as y = x³ − 4x or y = sin(x), the zeros, maximums and minimums are found from the graph itself, to 4 decimal places, in the part you can see. Drag or zoom and the list updates.

Example 1: where a parabola crosses the x-axis

For y = x² − 2x − 3, a = 1, b = −2 and c = −3. Then b² − 4ac = 4 + 12 = 16, which is positive, so there are two roots: x = (2 ± √16) ÷ 2 = (2 ± 4) ÷ 2, which gives x = −1 and x = 3. The vertex is at x = −(−2) ÷ 2 = 1, and y = 1 − 2 − 3 = −4 there, so the lowest point is (1, −4). Those are the numbers the calculator shows when it opens.

Example 2: where two graphs cross

The second line on the starting example is y = x + 1. The two graphs cross where x² − 2x − 3 = x + 1, which is x² − 3x − 4 = 0, or (x − 4)(x + 1) = 0. So x = 4 or x = −1. Put each back into y = x + 1 to get the points: (4, 5) and (−1, 0). The second one is on the x-axis, which is why it's also one of the parabola's roots.

Tables and lines of best fit

Press Add table to get two columns, x₁ and y₁. Type a value in each cell (Enter moves down, and a new row appears as you fill the last one) or paste two columns copied from a spreadsheet. The points appear on the graph (press ⤢ beside the graph to zoom to them; a paste does this for you), and the answer box gives the means and the correlation r. A second table gets x₂ and y₂.

To fit a formula to the points, type the y column, a tilde (~) and the formula, using the x column and any letters you want worked out: y₁ ~ mx₁ + b. The calculator picks the letters that make the total of the squared vertical misses as small as possible (least squares), draws the curve and gives R², the share of the spread in y the formula explains (1 is a perfect fit). The fitted letters can be used on other lines, so y = mx + b or 2m work afterwards. For a straight line it shows the textbook working:

slope m = (nΣxy − ΣxΣy) ÷ (nΣx² − (Σx)²) intercept b = (Σy − mΣx) ÷ n R² = 1 − Σ(y − ŷ)² ÷ Σ(y − ȳ)²

Example 3: hours of study and test scores

Five students studied for 1, 2, 3, 4 and 5 hours and scored 52, 60, 65, 74 and 79. The sums are n = 5, Σx = 15, Σy = 330, Σxy = 1,058 and Σx² = 55. So m = (5 × 1,058 − 15 × 330) ÷ (5 × 55 − 15²) = 340 ÷ 50 = 6.8, and b = (330 − 6.8 × 15) ÷ 5 = 45.6. The best-fit line is y = 6.8x + 45.6: about 6.8 more points for each extra hour. The misses add up to Σ(y − ŷ)² = 3.6 against a total spread of 466, so R² = 1 − 3.6 ÷ 466 = 0.9923, a very close fit. A line of best fit describes these five students; it doesn't promise that a sixth hour adds another 6.8 points.

Other shapes work the same way. y₁ ~ ax₁² + bx₁ + c fits a parabola, and y₁ ~ a·b^x₁ fits growth that doubles or halves at a steady rate. For curves like these the letters are found step by step from a starting value of 1, so a formula with a very different scale may need its letters to be close to 1 to settle.

Moving around the graph

Move: drag the graph with the mouse or a finger, or click it and use the arrow keys.
Zoom: scroll the mouse wheel, pinch on a phone, press the + and − buttons, or the + and − keys. Zooming centres on the mouse pointer.
Read a point: hover over a curve, or tap it, to see the coordinates of that point. Grey dots mark the intercepts, turning points and crossings of the line you're in.
Start again: ⌂ goes back to x from −10 to 10. Full screen gives the graph the whole window; Esc leaves it.
Settings: the wrench at the top right of the graph. It has larger text, reverse contrast (white on black), a square or polar grid, axis numbers, minor gridlines and arrows, a label, range and step for each axis, a lock for the view, Complex mode, and radians or degrees.

Both axes start on the same scale, so a circle looks round and a slope of 1 is a 45° line. Typing a range for x or y in the settings (−10 ≤ x ≤ 10) stretches just that axis; “Make x and y scales equal”, under More options, squares them up again. A step such as π/2 numbers the axis π/2, π, 3π/2 …, which suits graphs of sin and cos. The grid lines fall on 1, 2 or 5 times a power of ten, and the numbers on the axes stay at the edge of the graph when an axis is off screen.

Sliders and moving a graph

Sliders are the quickest way to see what each number in an equation does. Type y = a(x − h)² + k, press “add slider” for a, h and k, and drag them:

y = a(x − h)² + k a stretches the parabola (a < 0 flips it upside down) h moves it right (h > 0) or left k moves it up (k > 0) or down the vertex is (h, k)

The same works for any graph: y = f(x − h) + k shifts f right by h and up by k, and y = a·f(x) stretches it vertically by a.

Limits

The calculator draws real numbers only, so y = sqrt(x) stops at x = 0 and y = ln(x) has nothing to the left of the y-axis. A vertical jump, as in y = 1/x or y = tan(x), is left as a gap rather than joined up. Equations in both x and y are traced on a fine grid, which can miss a curve thinner than a few pixels; zoom in to check. Nothing you type is sent anywhere: the graph is drawn in your browser. For working out a single expression with every step, the scientific calculator is the better tool.

Frequently asked questions

How do I graph two equations at once?

Type each one on its own line. Every line gets its own colour, and the answer box lists where the line you're on crosses the others. With y = x² − 2x − 3 and y = x + 1 it gives the crossings (−1, 0) and (4, 5).

How do I find where a graph crosses the x-axis?

Click in its line: the answer box lists the x-intercepts, and grey dots mark them on the graph. For a straight line or a parabola they're worked out exactly; for anything else they're found to 4 decimal places in the part of the graph you can see, so zoom out if one might be off screen.

How do I make a slider?

Use a letter that isn't x or y, such as y = a(x − 2)², and press the “add slider: a” link that appears. You can also type a = 1 on a line of its own. Drag the slider and every graph that uses a moves with it.

Why doesn't sin(x) look right?

Check the angle setting at the bottom of the graph settings (the wrench at the top right of the graph). It starts in radians, so sin(x) repeats every 2π ≈ 6.28. In degrees it repeats every 360, so you'd need to zoom out a long way to see one wave.

How do I graph a circle?

Type it as an equation in x and y, such as x² + y² = 25. The answer box recognises a circle and gives its centre and radius: for (x − 2)² + (y + 1)² = 9 that's centre (2, −1), radius 3.

How do I find a line of best fit?

Press Add table and type your x values in the left column and y values in the right. Then type y₁ ~ mx₁ + b on a new line. The answer box gives m, b and R², and shows the sums the slope comes from; for 1 to 5 hours of study and scores of 52, 60, 65, 74 and 79 it gives y = 6.8x + 45.6.

Can I graph an inequality?

Yes. y > 2x − 1 shades the region above the line, and x² + y² ≤ 16 shades the inside of a circle. With < or > the edge is drawn dashed, because points on it aren't included; with ≤ or ≥ it's solid.

Graphing Calculator