Graphing Calculator
Graph functions, points and inequalities: sliders for parameters, roots, turning points and intersections listed, and pan and zoom on any device.
No points of interest in view
Type a function and it is drawn as you type: y = x² − 2, a circle x² + y² = 25, an inequality, a polar or parametric curve, or a list of points. Letters other than x and y become sliders, so a parabola's shape can be dragged into place.
Beside the graph, every root, y-intercept, turning point and crossing between rows is listed as text, not only marked as a dot. For lines and parabolas these points are exact: y = x² − 2 and y = x + 1 meet at ((1 − √13)/2, (3 − √13)/2) and ((1 + √13)/2, (3 + √13)/2). Nothing needs a sign-in.
How to use it
- Rows: each row is one expression, up to 50. Enter moves to the next row and adds one at the end. The coloured dot shows or hides a row; the cross removes it. A row that cannot be drawn says why under it, and the others stay on the graph.
- Sliders: a letter other than x and y, such as a in y = ax², is a parameter. While its row is typed it counts as 1; when you press Enter or leave the row, a slider row a = 1 is added, running from -10 to 10 in steps of 0.1. The range and step can be typed, and play moves it back and forth. Capital and small letters are different parameters.
- Polar and parametric rows take θ (or theta) and t, over a full turn unless you type another range under the row.
- Angles sets whether sin, cos, tan and θ read radians, the default, or degrees. Decimal places rounds the points that are found as decimals, the decimals given beside exact points that do not end, the trace and the table of values.
- The graph opens on x from −10 to 10, with both axes to the same scale. Drag to pan; zoom with the wheel, a pinch, the buttons, or + and − on the keyboard, where the arrow keys pan. The house button goes back to the opening view, and the last button shows the graph full screen where the browser allows it. Point at a function, or touch and drag along one, to trace it; tap a grey dot to show its coordinates. Pointing at a row of the points list or the table marks its point on the graph, and tapping it keeps the point marked.
- Table of values takes the place of the points of interest once there is a function, and lists any function row from a starting value in equal steps.
- Roots step by step, beside a parabola's points, opens the Quadratic Equation Solver with its a, b and c filled in, and Slope and line forms, beside a line's, opens the Slope Calculator with its equation, each worked in full. A row whose parameters are not typed as decimals has no such link.
- PNG and SVG download the graph as it is on screen.
How it works
Each function is sampled across the screen, and every stretch that bends more than half a pixel is sampled again, so curves stay smooth at any zoom. Where a function jumps, as tan x does at π/2, no line joins the two sides, and where it has no value, as √x for x below 0, the graph has a gap. Curves in x and y together, such as a circle, are traced across a fine grid. A strict inequality (< or >) draws its boundary dashed, since the line itself is not part of the region.
Points of interest are found for functions of x or y in view: where they cross the axes, their highest and lowest points, and where they meet each other or a curve in x and y. Crossings of a curve in x and y with the axes are found too. Where a row is a line or a parabola and its parameters are decimals, its points are worked exactly, as square roots where they do not end, with their decimals beside them. Every other point is found numerically, to the full precision of the computer's arithmetic, and shown to the decimal places chosen. At most 100 points are listed. Points on polar and parametric curves, and crossings of two curves in x and y, are not listed.
Examples
Where a parabola meets a line
- Row 1
- y = x^2 - 2
- Row 2
- y = x + 1
The parabola crosses the x-axis at (−√2, 0) and (√2, 0), and is lowest at (0, −2). The line crosses it at ((1 − √13)/2, (3 − √13)/2) and ((1 + √13)/2, (3 + √13)/2), the roots of x² − 2 = x + 1, worked exactly.
A parabola with sliders
- Row 1
- y = ax^2 + bx
- Row 2
- a = 0.5
- Row 3
- b = -1
With a = 0.5 and b = −1, the parabola crosses the x-axis at (0, 0) and (2, 0), and its vertex, the lowest point, is (1, −0.5). Drag a or b and the points follow; a negative a turns the parabola over, and the vertex becomes its highest point.
Inside a circle
- Row 1
- x^2 + y^2 <= 9
The disc of radius 3 is shaded, its edge solid because ≤ includes it. It crosses the axes at (−3, 0), (3, 0), (0, −3), (0, 3).
A crossing found numerically
- Row 1
- y = cos x
- Row 2
- y = x
cos x = x has no exact solution in closed form, so the crossing is found numerically: (0.74, 0.74) to 2 decimal places, in radians.
What you can type
| Row | Draws |
|---|---|
y = x^2 - 2 | A function of x |
x^2 - 2 | A function of x |
f(x) = 2^x | A function of x |
x = y^2 - 1 | A function of y |
y < x + 1 | A shaded region of x |
x^2 + y^2 = 25 | A curve in x and y |
x^2 + y^2 <= 9 | A shaded region in x and y |
r = 2sin(3θ) | A polar curve |
(cos t, sin t) | A parametric curve |
(1, 2), (3, 4) | Points |
a = 2 | A parameter's value, with a slider |
Frequently asked questions
See also
- Quadratic Equation Solver — for a parabola's roots and vertex with every step worked.
- Slope Calculator — for a line's slope, intercepts and equation from two points.
- Scientific Calculator — for a single value, with exact square roots and π.