Geometry Explorer Help

Using the Graph Menu:

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Using Geometry Explorer one can graph the relationship between two measured quantities.  The menu titled "Graph" has four groups of options which control graphing: Axes and Grid lines, Defining Functions, Adding Points to Graphs, and Miscellaneous Options.

Axes and Grid Lines:

(Show/Hide) Axes:  Coordinate axes are always available in Euclidean Geometry. However, the axes are hidden initially. To see the axes use this menu option. Once the axes are visible, the menu item will change to "Hide Axes", allowing one to make the axes invisible.

Grid (On/Off):  Grid lines are hidden initially. To see the grid lines use this menu option. Once the grid lines are visible, the menu item will change to "Grid Off", allowing one to make the grid lines invisible.

Here is an example with axes and grid lines both visible. Note that tic marks are shown on the axes to help identify coordinate values. Slightly longer tic marks denote multiples of the unit length (2,3,4,etc.). Also, note that an axes system is composed of two perpendicular lines, a point that serves as the origin, and a point on the positive x-axis that serves as the unit point, i.e. the point at (1,0) in this coordinate system.

axes

Defining Functions:

Add Function to Graph...: Use this option to add a function to the coordinate system.  Once the option is chosen, a dialog box will open up as shown.  You have the option of defining a standard y=f(x) function, a polar function (r = f(theta)), or a parametric function (h = (x(t), y(t)).  To input a function, type the name, the expression defining the function, and the minimum and maximum values for x (or theta or t, depending on the type of function), and hit Okay.

function panel

Here is what will be plotted in the main window:

y=f(x)

If we want to input a parametric function, we type it in using parentheses as shown in the dialog box below. Note that we can type in "pi" and the system will understand what we mean.

parameteric

Here is what will be plotted in the main window:

parametric example

Adding Points to Graphs:
Add As (x,y): We can add coordinate pairs to the coordinate system by selecting (in order) 1) the measurement that will serve as the x-coordinate and 2) the measurement that will serve as the y-coordinate.  In the figure below we have created a circle and now we want to graph the relationship between the radius of the circle and its area.  We select the radius distance (for x), then the area measurement (for y). 



At this point the menu item "Add As (x,y)" will be enabled. Once we click on this menu choice a point will be created at the (x,y) point corresponding to the measures selected and their relative distances on the axes in relation to the unit x-distance at point B. In the figure below we see that when the radius is about 1.5 the (x,y) point will be located at approximately 1.5 along the x-axis, and the y value will be at approximately 7.25 along the y-axis.  As we change the radius of the circle the measurements and the (x,y) point will update themselves accordingly.

 



Add Point on Function from x-Point: Use this option to add a new point to the graph of an existing function y=f(x).   For example, suppose we have constructed the graph of  f(x) = x^2
and have plotted a point A somewhere in the plane.  If we select the point and the graph, this menu item will become enabled and choosing this option will create a point B on the graph that will track the x-coordinate of B.

pt on function

Iterate function from Point...: Use this option to automate the iteration of a function on an initial point. In the figure below, we first constructed the graph of f(x) = x^2 and the point A. If we select A and the graph, this menu option will be enabled. Choosing this menu item will bring up a dialog box asking for the iteration level. We set this example at 2. Once we hit okay in the dialog box, the points B, C, D, and E will be created.  B is the image of the x-coordinate of A on the graph of f(x). C is the point on y=x (the green line) associated to B. Essentially, by plotting C we can use the y-value of B as the next x-value. Point D is then the f(x) value associated to the x-coordinate of C. Again, we move over to y=x yielding point E.  This process can be continued again and again, yielding f(x), f(f(x)), f(f(f(x))), etc.

iterated function

Misc Options:
Derivative of Function: Use this menu option to compute the derivative of a function. Select the graph of a function and choose this menu item. The graph of the derivative will then be plotted.

Input Box for Function: Use this menu option to create an onscreen textbox for changing the value of a function definition. Select the graph of a function to enable this option.  In the figure below we have created the graph of f(x) = x^2, and an input box for the function. At this point we could type in a new definition.

input box


Here we changed the definition to cos(x).

input box 2