standarddeviationcalculator.net

Updated Free · runs in your browser

Math

Graphing calculator

Type one to four functions of x and see them on the same axes. The calculator marks every zero and every point where two curves cross, and lists them with their coordinates.

Use x as the variable: ^ for powers, sqrt(x), abs(x), sin, cos, tan, exp, ln or log(x) (natural), log10, pi and e. 2x means 2*x.

Zeros · intersections 3 · 2
Zeros found3
Intersections found2
f₁(0), the y-intercept-4
f₂(0), the y-intercept-1
-5-4-3-2-1012345-10-505101520 (-2, 0)(2, 0)(0.5, 0)(-1, -3)(3, 5) x y

f₁(x) = x ^ 2 - 4   f₂(x) = 2 x - 1

PointWherexy
Zero (x-intercept)f₁-20
Zero (x-intercept)f₁20
Zero (x-intercept)f₂0.50
Intersectionf₁ = f₂-1-3
Intersectionf₁ = f₂35
Table of values
xf₁(x)f₂(x)
-521-11
-412-9
-35-7
-20-5
-1-3-3
0-4-1
1-31
203
355
4127
5219
Show the working, step by step

  1. Read each function. This is how the text was understood (2x means 2 × x):

    f₁(x) = x ^ 2 - 4 f₂(x) = 2 x - 1

  2. Sample every function at 2,001 evenly spaced x values from -5 to 5, and join the points. Gaps are left where a function is undefined (a square root of a negative, a division by zero).

  3. Find the zeros: wherever f(x) changes sign between two neighbouring samples, halve the interval repeatedly (bisection) until it is narrower than floating-point precision. A sign change at a vertical asymptote is thrown out, because f does not approach 0 there.

  4. Find the intersections the same way, by solving fᵢ(x) − fⱼ(x) = 0 for each pair of functions.

  5. Check the points by substituting x back in:

    f₁(-2) = 0 f₁(2) = 0 f₂(0.5) = 0 f₁ = f₂ at x = -1: y = -3 f₁ = f₂ at x = 3: y = 5

Points are found numerically, to about 10 significant figures. An exact answer such as √2 shows as 1.41421.

What the graph shows

The graph of y = f(x) is every point (x, f(x)). Three kinds of point are usually worth reading off it, and the calculator finds each one numerically:

zero (x-intercept): f(x) = 0 y-intercept: the point (0, f(0)) intersection: f(x) = g(x), i.e. f(x) − g(x) = 0

Finding intersections is the same problem as finding zeros, applied to the difference of two functions. That is how a graph solves equations: to solve x² − 4 = 2x − 1, plot both sides and read the x-coordinates where they cross.

A worked example

The calculator opens with f₁(x) = x² − 4 and f₂(x) = 2x − 1 on −5 ≤ x ≤ 5.

  1. Zeros of f₁: x² − 4 = 0 gives x = −2 and x = 2.
  2. Zero of f₂: 2x − 1 = 0 gives x = 0.5.
  3. Intersections: x² − 4 = 2x − 1 rearranges to x² − 2x − 3 = 0, which factorises as (x − 3)(x + 1) = 0. So x = 3, where y = 2(3) − 1 = 5, and x = −1, where y = 2(−1) − 1 = −3.
  4. y-intercepts: f₁(0) = −4 and f₂(0) = −1.

The graph marks the five points (−2, 0), (2, 0), (0.5, 0), (−1, −3) and (3, 5). The headline reads “3 · 2”: three zeros and two intersections.

Functions you can use

TypeExampleNotes
Polynomialx^3 - 3x + 12x means 2*x
Rational(x + 1)/(x - 2)Break at the asymptote x = 2
Rootsqrt(x), nthRoot(x, 3)sqrt is undefined for x < 0
Exponentialexp(x), 2^x, e^(-x)
Logarithmln(x), log10(x), log(x, 2)Defined for x > 0
Trigonometricsin(x), cos(2x), tan(x)Radians
Absolute valueabs(x - 1)

Choosing a good window

Most of the trouble with graphs is the window. If a parabola looks like a straight line, the x range is too narrow; if it looks like a thin V, the y range is too tall. Start with the automatic y range, then set it by hand once you know where the interesting part is. For trigonometric functions, a range such as −7 to 7 shows just over two full periods of sin x. For exponentials, keep the range modest: ex at x = 20 is already about 485 million.

Accuracy

The points are found by bisection, which halves the search interval until it reaches the limit of double-precision arithmetic, so zeros and intersections are accurate to roughly ten significant figures. An exact value such as √2 is shown as 1.41421. For an exact answer to a polynomial equation, use the quadratic equation calculator or the polynomial calculator.

Common questions

How do I type functions into the graphing calculator?

Use x as the variable and write the right-hand side of y = …: x^2 - 4, 2x - 1, sin(x)/x, sqrt(x + 3), exp(-x^2). A leading y = or f(x) = is ignored. ^ is a power, 2x means 2 × x, ln and log are both the natural logarithm, log10 is base 10, and pi and e are the constants. Trig functions use radians.

How does it find where two graphs intersect?

Two graphs meet where f(x) = g(x), which is where the difference f(x) − g(x) is zero. The calculator samples the difference at 2,001 points across the x range, finds each place where it changes sign, and narrows each one down by bisection to about ten significant figures. For x² − 4 and 2x − 1 it finds x = −1 and x = 3, the solutions of x² − 2x − 3 = 0.

Why does it miss some zeros?

Only zeros inside the x range are searched, so widen it if you expect roots further out. Two zeros closer together than the sampling gap (the range divided by 2,000) can hide between samples. A root where the curve touches the axis without crossing, as x² does at 0, is caught by a separate check for a minimum of |f| at zero.

What happens at a vertical asymptote?

For a function like 1/x or tan x, the values jump from large negative to large positive across the asymptote. That is a sign change, but not a zero, and the calculator rejects it because |f| does not get small there. The curve is also broken at the jump rather than joined with a vertical line.

Can I set the y range myself?

Yes. Leave both y boxes blank for an automatic range, which ignores the most extreme 2% of values at each end so an asymptote does not flatten the rest of the graph. Fill in both to fix the window, for example −10 to 10 to compare several curves on the same scale.

Written and reviewed by our editorial team. Last updated . Method and sources: how these numbers are computed.