The GraphFunc tool (beta version) is a program online that is used to plot the graph of mathematical functions in two or three coordinate systems and help exploring several topics in calculus.  It has the following features:

• Plotting the elementary function(s) in two or three dimensional Cartesian, Polar and Parametric coordinates (see demo).
• Computing the first derivative and second derivative of f(x) in Cartesian coordinates.
• Computing the values t, x, f(t), f(x) and the first and second derivatives of these functions in Polar or Parametric coordinates.
• Computing the extreme value(s) or the maxima and minima of f(x) (see demo).
• Computing the x-intercepts where f(x) = 0.
• Computing the area under a curve (see demo).
• Computing the surface area of a solid of revolution.
• Computing the arc length of a curve between two points (see demo).
• Computing the volume of a solid of revolution with the disc and washer methods.
• Finding the equations of tangent and normal to the curves and plotting them.
• Solving a System of Linear Equations.
• Solving an equation in one variable.
• Computing the sum of a series.
• Supporting Mortgage Calculator.
• Computing a value of Gamma Function.
• Dragging and rotating the view in 2D and 3D.
• Supporting zoom in and out of a graph.

When you access this website or click on a GraphFunc hyperlink in this page, the GraphFunc tool (applet) will be displayed in the popup window that you can use to graph your functions online.  If your browser does not display GraphFunc, check whether you need to download JRE (Java Runtime Environment) from Sun or setup your browser security in secure default configuration.  (Try theversion if your computer has JRE 1.5x or 1.6x.)

How to use GraphFunc tool

Use "Graph It!" Button To Plot A Graph
Enter a function f(x) in the text input (or command line) marked “f(x) = .  Then click on the Graph It! button to draw the graph in 2D Cartesian coordinates (by default).  The graph can be moved or dragged to any positions.  Always click on the "Graph It!" button each time a new or modified expression is given in order to have it effective before using any other buttons.

Cartesian, Polar or Parametric Coordinates Mode (see demo)

The option under the Cartesian dropdown list box helps User to plot the function(s) in Polar or Parametric coordinates.   Use x as the input variable of f(x) in two-dimensional Cartesian coordinates, and x, y as the variables of z = f(x, y) in three-dimensional Cartesian coordinates.  For the Polar and Parametric co-ordinates, t is used as the input variable of f(t).

Single/Multiple Graph - Plot Multiple Graphs Synchrnous

Want to plot two or more graphs synchronous.  Need to switch to Multiple Graph mode by selecting the Multiple Graph item from the Single Graph drop-down list box.  After switching to Multiple Graph mode, a list of the text inputs is displayed.  Then enter any functions in the text inputs and click on the Graph It button to see its graphs.  A current active function is indicated by a red indicator marked on f1(x) = , f2(x) = , ...or f5(x) =.  To turn a function to be an active, click with the mouse on either a text field or its function label.  For example, if there are five functions entered in text fields and the user wants to compute the area under the curve f3(x). First, click on the text field marked "f3(x) = " to make sure the red indictor is displayed.  Second, provide the values of the lower and upper bounds then click on the Find Area button to get the result.  The behavior of this example can be applied to other computing functionalies such as volume, tangent, extremum, x-intercepts, and derivatives.

Calc Buttons - Compute f and Its First & Second Derivative at a Given Point

The values of f(x1), f '(x1) (first derivative) and f "(x1) (second derivative) at a given point x1 can be computed by clicking on the Calc button in Cartesian coordinates.  These values can be also determined when clicking with the mouse at any points on the graph.   This feature is applied to the Polar and Parametric coordinates in which its variable is denoted "t".   When in either Polar or Parametric coordinates mode, there are two Calc buttons: one is used for compution in terms of variable t; the other is used for computation in terms of variable x.

Area Button - Compute Area Under A Curve (click here to see a demo)
The area under a graph in 2D (not 3D) Cartesian coordinates is computed by clicking on the Find Area button right after the values of the lower (label marked "From") and upper bounds (label marked "To") are provided.

Length Button (click here to see a demo)

To compute the length of the arc of f(x) between two points along a curve, enter the "From" and "To" values on its text fields, then click on the Find Length button to get the result.

Maxima or Minima (Extremum)
To find a local maxima or minima of function f(x), select the Extremum item from the Functions drop-down list box.  Then click on the graph at any points near to the vertices to get the computed results.  If a maxima or a minima value is found when clicked, it shows the computed results onto the text field at the label marked x = and f(x) = .

X-Intercept

To find the x-intercepts where f(x) or y = 0, choose the X-intercept item from the Functions dropdown list box, and then click on the graph at some points where f(x) nearly crosses the x-axis.  If a value of x as f(x) approaches to 0 is found, its result is computed and displayd onto the labels marked x = and f(x) =.  This option also implies to find the roots of the equation when f(x) = 0.

Tangent/Normal
To find the tangent or normal line to f(x) at a given point x, click on the Tangent button to display the Add Tangent/Normal dialog.  Enter a value of x and then click on the OK button to see the result.  In addition, the dynamic tangent line is shown while clicking with the mouse on any points on the graph or dragging it along the horizontal x-axis.

Sum Series

GraphFunc has a capability of computing the sum of a series up to nth terms.  To use this functionality, select the Sum Series item from the Functions dropdown list to display the Numerical Computation of Series dialog.  From this popup dialog enter a series expression in the command line, and provide the values in the textfields with labels marked Start, End and in steps of , which represent the first term, the last term and the step-width of the series, respectively.  Afterwards, click on the Compute Sum button to get the sum.

System of Linear Equations
Want to solve a
System of Linear Equations with n equations and n unknowns.  Select the Linear Equations item from the Functions dropdown list, a window with four equations and four unknowns (by default) is displayed.  From this window enter the coefficient values of equations in the text fields then click on the Solve button to get the result.

View 2D

The View 2D dropdown list box is used to switch the graph(s) from 2D to 3D or vice versa.  The graph(s) can be plotted in Cartesian, Polar or Parametric coordinates by selecting an item under Cartesian dropdown list box.  Currently, GraphFunc supports two variables x and y as variables of function z = f(x,y) in three-dimensional Cartesian coordinates.  Use "t" as the input variable of Polar function r(t) or Parametric functions x(t), y(t) in the Polar or Parametric coordinates.

View 3D

Select the View 3D mode, it displays a graph in the three dimensional coordinates in which its motion by default is shown.  Click on the Motion checkbox to turn it on or off.

Volume or Surface Area Button

Select the x-axis item from the Rotation dropdown list box to rotate the graph around the x-axis.  Click on the Volume or Surface Area button to compute the volume or the surface area generated from this revolution after the values of the lower bound (label marked "From") and the upper bound (label marked "To") are provided.

Gamma Computation - Compute a Value of Gamma Function
Select Gamma Function item from Math Utilities menu.  Enter a value for x and then click on the Gamma Computation button to get the result.

Zoom In / Out Button
A graph can be zoomed in or out when clicking on the Zoom In or Zoom Out button.

Reset Button

Clicking on the Reset button is to clear data and reset the GraphFunc tool's configuration to its default values.

(Note: The below instructions are applied to the new and old GraphFunc versions.  However, some are not applied to the old version.)

Some examples

 Two-dimensional plotting functions Enter f(x) = xsin(x) x*sin(x)  (enter this function into command line) x^3 +x -1 (x^3 + x + 4)/(x - 1) (sin(x))^3 + cos(x) Three-dimensional plotting functions Enter f(x,y) = z = f(x,y) = sin(x+y) sin(x+y) x^2 + y^2  - 3 exp(x) + sin(y)  or e^x + sin(y) Polar Graphs (2D) Select the Polar item from the Cartesian dropdown listbox, then enter 6/(6 - 5*sin(t)) -sin(5*t) (Butterfly Curve) e^(cos(t)) - 2*cos(4*t) + sin(t/12)^5 Parametric Graphs  (3D) Select the Parametric item from the Cartesian dropdown list, then enter sin(t) + 2*cos(t)      (into the text field with label marked f1(t) = ) sin(2*t)                   (into the text field with label marked f2(t) = ) cos(t) sin(t)

(More functions and examples...)

Now choose a function and use GraphFunc tool as shown above or click here to graph it.

 My fellow Americans: ask not what your country can do for you – ask what you can do for your country. John F. Kennedy

Thank you for using GraphFunc online.  Any suggestions and comments are welcome to sms_century21@seriesmathstudy.com .

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