Graphing Calculator

Plot up to three functions of x, with a parameter slider, a tangent line and implicit curves, and find the roots, turning points, y-intercepts and every point where the graphs intersect.

y₁ =	x^2 - 4
y₂ = (optional)	2*x + 1
y₃ = (optional)
x from	-6
x to	6
Parameter a (use a in your functions)	1
Tangent to y₁ at x = (optional)
Implicit curve F(x, y) = G(x, y) (optional)

y₁ =

y₂ = (optional)

y₃ = (optional)

x from

x to

Parameter a (use a in your functions)

Write a in y₁, y₂ or y₃ (like a x^2) and drag the slider to see the graph change.

Tangent to y₁ at x = (optional)

Implicit curve F(x, y) = G(x, y) (optional)

For example x^2 + y^2 = 16 or x^3 + y^3 = 6x y.

Roots of y₁

−2, 2

y₁ crosses the y−axis at −4. y₁ and y₂ meet at (−1.4495, −1.899), (3.4495, 7.899).

Approximate. Not independently checked.

Method: Sign changes refined by bisection; turning points from the exact derivative
Worked out with: JavaScript floating point

Graph

y₁ root x = -2at (-2, 0)Point at the chart, or Tab to it and use ← →

y = x^2 - 4y = 2*x + 1y₁ root x = -2: (-2, 0)y₁ root x = 2: (2, 0)y₁ min (0, -4): (0, -4)y₂ root x = -0.5: (-0.5, 0)y₁ ∩ y₂ (-1.449, -1.899): (-1.45, -1.9)y₁ ∩ y₂ (3.449, 7.899): (3.45, 7.9)

Key points (6 significant figures)

Graph	Point	Where
y₁	Root (crosses or touches y = 0)	x = −2
y₁	Root (crosses or touches y = 0)	x = 2
y₁	Local minimum	(0, −4)
y₁	y-intercept	(0, −4)
y₂	Root (crosses or touches y = 0)	x = −0.5
y₂	y-intercept	(0, 1)
y₁ and y₂	Intersection	(−1.44949, −1.89898)
y₁ and y₂	Intersection	(3.44949, 7.89898)

About the Graphing Calculator

Type functions with x as the variable, e.g. x^2 - 4, 2*sin(x), sqrt(x), ln(x) or e^x. Supported functions include sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, ln, log, sqrt, cbrt, exp and abs; angles are in radians and pi and e are available. Roots and intersections are found numerically, from sign changes refined by bisection, and roots where the graph only touches the axis (like x² at 0) are found too. Turning points are where the exact derivative is zero and the graph changes direction. All values are shown to 6 significant figures.

The formula

Roots: values of x where f(x) = 0; intersections: f₁(x) = f₂(x)

Worked example

Example: Roots of y₁ = −1, 3

Inputs: y₁ = x^2 - 2x - 3; y₂ = (optional): x + 1; x from: -6; x to: 6; Parameter a (use a in your functions): 1.
Formula: Roots: values of x where f(x) = 0; intersections: f₁(x) = f₂(x)
Result: Roots of y₁ = −1, 3

Questions people ask

How do I write powers and multiplication?

Use ^ for powers and * for multiplication (2*x), though 2x also works.

Why is part of the curve missing?

The function is undefined there (e.g. sqrt of a negative) or very large.