Graphing Calculator
Plot up to 6 graphs and explore them: drag, pinch or scroll to zoom, tap a curve to trace it, and get every root, turning point and intersection listed for you.
Tap or hover on a curve to trace it.
Drag to move. Pinch, scroll or use + and − to zoom. Fit brings everything back into view.
t is for parametric curves (x(t), y(t)) and θ for polar curves r = …. You can type π, like 2π or π/2. In degrees mode use 360.
How your graph is worked out
Graph 1, step by step: read it, plot lots of points, then search them for roots and turning points.
- 1
Read your line
y = x² − 2x − 3one y for each xkind: y = f(x)
- 2
Plot lots of points
x = 0 → y = −3x from −3.5 to 6.51,500 steps of 0.0067
- 3
Find where y = 0
sign flips, halve the gapx = −1, x = 32 roots in view
- 4
Find the turns
slope changes signmin (1, −4)1 turning point
Read more: the formulas, a worked example and what you can type
A graph of y = f(x) shows, for every x across the page, how high y is. This calculator draws it by working out y at hundreds of x values and joining the dots, then looks along the curve for the points people usually want: roots (where it crosses the x-axis), turning points (the tops and bottoms of hills and valleys), the y-intercept and the places two graphs cross.
The formula
root: f(x) = 0, found where y changes sign, then the gap is halved until it is tiny
turning point: slope f′(x) = 0; minimum if the slope goes − to +, maximum if + to −
y-intercept: (0, f(0))
intersection of f and g: f(x) − g(x) = 0, then y = f(x)
area from p to q: ∫ f(x) dx by Simpson’s rule, h ÷ 3 × (y₀ + 4y₁ + 2y₂ + 4y₃ + … + yₙ)
parametric: (x, y) = (x(t), y(t)); polar: x = r cos θ, y = r sin θ
Worked example
Graph 1 is y = x² − 2x − 3 and graph 2 is y = x + 1. Roots of graph 1: at x = 2 it is 4 − 4 − 3 = −3 and at x = 4 it is 16 − 8 − 3 = 5, so it crosses 0 in between. Halving the gap again and again lands on x = 3, and the same on the left gives x = −1. Check: it factors as (x + 1)(x − 3). Turning point: the slope is 2x − 2, which is 0 at x = 1, where y = 1 − 2 − 3 = −4. The slope goes from negative to positive, so (1, −4) is a minimum. y-intercept: at x = 0, y = −3. Intersections: x² − 2x − 3 = x + 1 means x² − 3x − 4 = 0, which is (x − 4)(x + 1) = 0, so x = 4 or x = −1. Put them into y = x + 1: the graphs meet at (4, 5) and (−1, 0). Area under graph 1 from x = −1 to x = 3: [x³ ÷ 3 − x² − 3x] gives (9 − 9 − 9) − (−1/3 − 1 + 3) = −9 − 5/3 = −32/3 ≈ −10.6667. It is negative because that part of the curve is below the x-axis.
What you can type
| Type | Means | Example |
|---|---|---|
| x^2, x², x^(1/3) | Powers and roots | x^3 − 3x |
| 2x, 3(x + 1), (x + 1)(x − 2) | Times, without the × | 2x(x − 1) |
| sqrt(x), √x, cbrt(x), abs(x), |x| | Square root, cube root, size | |x| − 2 |
| sin, cos, tan, sec, csc, cot, asin, acos, atan | Trig (radians or degrees) | sin 2x, sin^2(x) |
| ln, log, log2, exp, e, π | Logs (log is base 10) and constants | e^(−x^2) |
| floor, ceil, round, sign, min, max, mod, ! | Steps, limits, factorial | floor(x) |
| y > …, y ≤ … | Inequality, shaded | y > x^2 − 4 |
| x = 3, x < 1 | Vertical line or band | x = −2 |
| (x(t), y(t)) | Parametric curve | (3cos(t), 2sin(t)) |
| r = …θ… | Polar curve | r = 2sin(3θ) |
| a, b, c | Sliders | a(x − b)^2 + c |
Good to know
- Points are found in the part of the graph you can see. Drag or zoom out to find roots further away; the list updates.
- Jumps, like tan x at 90° or 1/x at 0, are not counted as roots or turning points even though y changes sign there.
- Answers are found with numbers, not algebra, and are shown to 4 decimal places. Whole numbers and simple multiples of π are recognised, so a root at 3.14159… shows as 3.1416 (π).
- A root where the curve only touches the axis, like y = x² at 0, is found by looking for where y gets closest to 0.
- Equations that mix x and y, like x² + y² = 4, can be drawn as a parametric curve: (2cos(t), 2sin(t)).
Everything is worked out in your browser. Nothing you type is sent anywhere.
Common questions
How do I graph a function?
Type the right-hand side of y = … in a box, like x^2 − 2x − 3, and it is drawn straight away. You can also type the whole thing, y = x^2 − 2x − 3 or f(x) = …. Use ^ for powers, sqrt or √ for square roots and * or nothing for times, so 2x means 2 × x. Add up to 6 graphs and tap a coloured dot to hide one.
How do I find where two graphs intersect?
Type both graphs. Every place they cross in the part you can see is listed under Intersections. The calculator subtracts one from the other and finds where the difference changes sign. For y = x² − 2x − 3 and y = x + 1 that is x² − 3x − 4 = 0, so they meet at (−1, 0) and (4, 5). Move or zoom the graph to look further out.
How do I find the roots and turning points of a graph?
They are listed for every y = f(x) graph. Roots (x-intercepts) are where y = 0: for y = x² − 2x − 3 they are x = −1 and x = 3. A turning point is a minimum or maximum, where the slope changes sign: here a minimum at (1, −4). The y-intercept is the value at x = 0, which is −3. Tap any point to see it on the graph.
Can I plot parametric, polar and inequality graphs?
Yes, in Advanced. Type a pair like (cos(t), sin(t)) for a parametric curve, r = 1 + cos(θ) for a polar curve, or y > x^2 − 4 for an inequality, which shades every point above the curve. A dashed edge means the edge itself is not included (> or <). x = 3 draws a vertical line. You can set the t and θ ranges.
Should I use degrees or radians?
Radians are the default, as in most maths: sin(x) repeats every 2π, about 6.2832, and roots of sin x show as 3.1416 (π). Pick Degrees in Advanced to make sin(x) repeat every 360, so its roots are 0, 180 and 360. The choice applies to every trig function on the page and to θ in polar graphs.
What are the sliders a, b and c for?
Use the letters a, b or c in a graph, like a·sin(b·x) + c, and a slider appears for each. Drag it, type a value or press play to watch how the graph changes. They start at a = 1, b = 1 and c = 0. This is a quick way to see what each number in an equation does, such as how a stretches and c shifts.
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