OneTaskBy Tim King · Updated
Function Grapher
Roots, Turning Points, Y-Intercept and Curve Crossings Marked on the Graph
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How the Function Grapher works
This function grapher plots up to three functions of x on the same axes. Type something like x^2 - 2, sin(x) or 1/x and the curve appears straight away. The grapher finds the roots, the y-intercept, every local minimum and maximum in view and the points where two curves cross, marks them on the graph and lists them beside it. Hover or touch the graph to read off values.
A graph turns a formula into a shape. Where the curve crosses the x-axis the function is zero, which is a root. Where it turns from rising to falling it has a maximum, and the other way round a minimum. Reading those features off a picture is far quicker than solving for them by hand, and seeing two curves together shows at a glance where they are equal.
How the function grapher works
The grapher evaluates each function at well over a thousand points across the x range. A change of sign between two neighbouring points means a root lies between them, and the grapher narrows it down by repeated halving to many decimal places. Turning points are found the same way from the shape of the curve, then refined. Intersections are the roots of the difference between two functions. When the y range is left blank it is chosen from the middle 96% of values, so a single spike near an asymptote does not squash the rest of the graph flat.
You can write powers with ^, multiply by writing terms side by side (3x, 2sin(x)), and use sin, cos, tan, sqrt, abs, ln, log (base 10) and exp, with pi and e as constants.
Tips
- Zoom with the limits. Narrow the x range around a feature to see it in more detail.
- Fix the y range when an asymptote makes the auto scale jump about, as with tan(x) or 1/x.
- Solve an equation by graphing. To solve x² = 2 + sin(x), plot both sides and read off where they meet.
FAQ
How do you find the roots of a function from its graph?
The roots are where the curve crosses or touches the x-axis, because y is zero there. The grapher lists them in order. For x^2 - 2 they are about −1.414 and 1.414, which are −√2 and √2.
Why is part of my graph missing?
The function has no real value there. sqrt(x) stops at x = 0, ln(x) only exists for positive x, and 1/x breaks at x = 0. The grapher leaves a gap rather than drawing a false line across it.
How do I graph trig functions in degrees?
The grapher works in radians, the standard for graphs. To use degrees, convert inside the function: sin(x pi/180) plots sine with x in degrees, so set the x range from 0 to 360.
Can I graph a vertical line or a circle?
Not directly, because those are not functions of x. A circle of radius 3 can be drawn as two halves: sqrt(9 - x^2) and -sqrt(9 - x^2), one in each of the first two boxes.
See also: all Trigonometry and Calculus tools, Arithmetic, Number Theory, Algebra and Geometry, Statistics.
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