Graphing Calculator | Function, Equation, Parametric, Point

About the Graphing Calculator

Our online graphing calculator is a sophisticated, feature-rich, and user-friendly tool for graphing functions, equations, parametric curves, or points in the user's chosen Cartesian or polar coordinate system.

The graphing calculator also effortlessly determines x-intercepts (also known as zeros or roots) of a function and computes symbolic derivatives up to the second order for both functions and parametric equations.

Unique Features of the Graphing Calculator

To enhance the understanding of how graphs of functions are created in the polar coordinate system, as well as graphs of parametric equations in both Cartesian and polar coordinate systems, our graphing calculator utilizes a sophisticated and unique interactive animation method to progressively draw these types of graphs automatically, step by step. This is a tremendously useful feature for visualizing their graphs as they are being plotted. It also offers full control over animation, allowing users to run, pause, resume, and adjust the animation speed through a convenient and intuitive interface.

It is also unique in its ability to visualize graphs in a parallelogrammatic coordinate system (oblique coordinate systems—non-orthogonal Cartesian and generalized polar coordinate systems)—where axes can be rotated and intersect at any angle. These features provide an interactive way to explore and understand graphing in non-standard coordinate systems.

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Functions

Lines

1 x+1 2x

Semi-circles

√(9-x^2) -√(9-x^2)

Semi-ellipses

√(9-x^2/3) √(9-x^2/3)

Parabolas

x^2 0.5x^2-4x+1 -(0.5x^2-4x+1)

Semi-hyperbolas

√(x^2-4) -√(x^2-4)

Other graphs

√(4sin(2x)) √(4cos(2x))

Functions – Polar

Lines

2csc(θ) 2sec(θ) 1/(sin(θ) - cos(θ))

Circles

1 2 6sin(θ) 8cos(θ)

Spirals

θ θ/5 dom=(0, 10π) √(θ) dom=(0, 10π) 1/θ dom=(0, 10π)

Roses

4sin(3θ) 4sin(2θ) 4sin(5θ) 4sin(4θ)

Ellipses

1/(1-.8cos(θ)) 1/(1-.8sin(θ)) 1/(1+.8cos(θ)) 1/(1+.8sin(θ))

Parabolas

1/(1-sin(θ)) 1/(1+cos(θ)) 1/(1+sin(θ)) 1/(1-cos(θ))

Hyperbolas

1/(1+2cos(θ)) 4/(1+2sin(θ)) 1/(1-2cos(θ)) 4/(1-2sin(θ))

Cardioids

3+3cos(θ) 2+2sin(θ) 3-3cos(θ) 2-2sin(θ)

Limacons

2+3cos(θ) 1+2sin(θ) 2-3cos(θ) 1-2sin(θ)

Lemniscates

√(4sin(2θ)) √(4cos(2θ))

Butterfly curve

e^sin(θ)-2cos(4θ)+sin((2θ-π)/24)^5 dom=(0, 12π)

Equations

Lines

y = 1 x = 1 y = x+1 x = y+1 3x + y = 2 3x - y +5 = 4x+2y-2

Circles

x^2+y^2 = 9 (x-2)^2 + (y-2)^2 = 4

Ellipses

x^2/4 + y^2/9 = 1 x^2-xy+2y^2-x-2y-8=0

Parabolas

y=x^2 y = x^2-4x+4 2x^2-4xy+2y^2-x-2y-2=0

Hyperbolas

x^2/4 - y^2/9 = 1 2x^2-5xy-4y^2+9x+9y-16=0

Other graphs

x^2 = y^2 sin(xy) = cos(xy)

Equations — Polar

Currently, not available.

Parametric

Lines

[t, 1] dom=(-5, 5) [1,t] dom=(-5, 5) [t, 2t] dom=(-5, 5)

Circles

[4sin(t), 4cos(t)] [3sin(t)+1, 3cos(t)+1]

Ellipses

[4cos(t), 3sin(t)] [3cos(t), 4sin(t)] [4sin(t), 3cos(t)] [3sin(t), 4cos(t)]

Parabolas

[t, t^2] dom=(-4, 4) [t^2, t] dom=(-4, 4)

Hyperbolas

[3sec(t), 4tan(t)] [3tan(t), 4sec(t)]

Other parametric graphs

[5sin(t), 4cos(t)] [5sin(t), 4cos(2t)] [5sin(t), 4cos(3t)] [5sin(2t), 4cos(t)] [5sin(2t), 4cos(3t)] [5sin(2t), 4cos(5t)] [5sin(3t), 4cos(5t)] [5sin(3t), 4cos(7t)] [5sin(5t), 4cos(7t)] [5sin(7t), 4cos(9t)]

Butterfly curve

[sin(t)(e^cos(t)-2cos(4t)-sin(t/12)^5), cos(t)(e^cos(t)-2cos(4t)-sin( t/12 )^5)] dom=(0, 12π)

Parametric – Polar

Lines

[2csc(t), t] [2sec(t), t] [1/(sin(t) - cos(t)), t]

Circles

[1, t] [2, t] [6sin(t), t] [8cos(t), t]

Spirals

[t, t] [t/5, t] dom=(0, 10π) [√(t), t] dom=(0, 10π) [1/t, t] dom=(0, 10π)

Roses

[4sin(3t), t] [4sin(2t), t] [4sin(5t), t] [4sin(4t), t]

Ellipses

[1/(1-.8cos(t)), t] [1/(1-.8sin(t)), t] [1/(1+.8cos(t)), t] [1/(1+.8sin(t)), t]

Parabolas

[1/(1-sin(t)), t] [1/(1+cos(t)), t] [1/(1+sin(t)), t] [1/(1-cos(t)), t]

Hyperbolas

[1/(1+2cos(t)), t] [4/(1+2sin(t)), t] [1/(1-2cos(t)), t] [4/(1-2sin(t)), t]

Cardioids

[3+3cos(t), t] [2+2sin(t), t] [3-3cos(t), t] [2-2sin(t), t]

Limacons

[2+3cos(t), t] [1+2sin(t), t] [2-3cos(t), t] [1-2sin(t), t]

Lemniscates

[√(4sin(2t)), t] [√(4cos(2t)), t]

Other parametric graphs

Entering Expressions into Graphing Calculator

The graphing calculator is designed to be intelligent and user-friendly. When you enter an expression, it detects its type and internally adjusts the variables accordingly (to see the adjustments, just hover your mouse over the expression type label above the relevant input box):

Function: If you use x as the independent variable, the function graphing calculator will label the expression as f(x)= or f(θ)= depending on the coordinate system.

Equation: If your expression includes an equal sign (=), the calculator switches to equation mode, and the equation graphing calculator (also known as implicit function grapher) will display Eq: in the label.

Points: If your expression uses both commas and semicolons, the calculator switches to points mode, and the points graphing calculator will display x,y; or r,θ;, depending on the coordinate system, for the expression label.

Parametric curve: A single comma indicates a parametric expression, switching the calculator to parametric mode. The parametric curve graphing calculator (also known as parametric equations grapher) displays p(t)= for the expression label, and replaces x with t in the expression. Deleting the comma switches back to function mode, replacing t with the appropriate variable.

Note: When graphing functions or parametric expressions, if you don't specify a domain (interval), the graphing calculator automatically selects a suitable domain for accurate plotting. The default domains are:

dom=(-∞,∞) for graphing functions in the Cartesian coordinate system.

dom=(0,2π) for graphing functions in the polar coordinate system.

dom=(0,2π) for graphing parametric curves in both Cartesian and polar coordinate systems.

Users can modify the endpoints of the interval as needed. However, for polar or parametric graphing, the endpoints must be finite. If infinite values are entered, the calculator automatically adjusts them to finite values.

Specific Applications of the Graphing Calculator

Beyond general expression, this graphing calculator allows you to:

Graph linear functions and linear equations in point-slope form and slope-intercept form.

Graph quadratic equations representing conic sections in the standard form (x-h)² + (y-k)² = r² and the general form Ax² + Bxy + Cy² - Dx + Ey + F = 0 which may represent a circle, ellipse, parabola, hyperbola, or some degenerate graphs.

Graph level curves, expressed as F(x,y) = c.