Tutorial I. A Beginner’s Guided Tour

• For a general description for the GrafEq program, visit the on-line description page at http://www.peda.com/grafeq/.
• For an overview of the program’s functionalities, visit the on-line specifications page at http://www.peda.com/grafeq/.
• For a complete reference, consult the GrafEq manual.

• This tutorial is based on GrafEq 2.05; users of other versions of GrafEq might need to adjust some steps for any differences in the user interface.

To launch the program

1. Double-click on the GrafEq icon, as seen in figure I.1 following:
   
   Figure I.1: The GrafEq icon
   

2. A splash screen appears, which gradually reveals the GrafEq title screen, as seen in figure I.2 following. Press the space bar once to clarify the splash screen, and once more to remove it.
   
   Figure I.2: The GrafEq logo
   

3. The Relation #1 algebraic window then opens automatically, as seen in figure I.3 following:
   
   Figure I.3: An Algebraic Relation Window
   

To enter a relation

1. Ensure that the caps lock key is not depressed.
   
   
2. Follow the step sequence, which will be provided, to enter the relation as seen in figure I.4 following:
   
   Figure I.4: y=sqrt(x)sin1/x
   

   The necessary step sequence is as follows (Refer to the hints immediately following regarding easy buttons and prompts.):

   1. Type in “y=”;
   2. Click the sqrt() easy button;
   3. Type in “x”, then “)”, noting the prompt at the bottom of the relation window, that says “Press ) to finish the term”;
   4. Type in “sin1/x”.
   Some general hints about entering relation specifications:
   • Use the keyboard to enter letters, digits, brackets, and simple symbols such as +, -, *, /, !, >, <, =, ...
   • Use easy buttons to enter special characters and operators, such as algebraic powers and roots. Figure I.5 following shows the easy button floating window:
     
     Figure I.5: Easy Button Floating Window
     

   • The relation window also displays a prompt message to guide the user. In an empty relation window, as in figure I.5, the help message prompts the user to enter a relation. Other appropriate messages appear at different stages of relation input.
   • The relation window parses entries as a relation specification is being entered, and displays them in standard mathematics format. The sequence of keyboard keys pressed can be revealed by moving the cursor from the end of the entry line to the left.
   • To make changes to the relation, or correct typing mistakes:
     1. position the cursor, with the mouse or the left and right arrow keys, to where changes are to be made; (Any relation portions now behind the cursor would become unformatted, as a sequence of keyboard entries; don’t worry about them, they will become formatted again when relation entry is completed, or when the cursor is placed back to the end of the entry line.)
     2. press the delete key to erase the character to the left of the cursor, and
        press the del key to erase the character to the right of the cursor;
   
   
3. The prompt message at the bottom of the window now states “Press return to graph...” Press return now. The Create View window will then appear. Figure I.6 following is the create view window:
   
   Figure I.6: The Create View Window
   

To specify a viewport

1. GrafEq can plot both Cartesian and Polar graphs, and will automatically set the default mode according to the relation entered. The graph mode can be changed by clicking its radial button. Since we have entered Relation #1 as a rectangular relation, Cartesian mode is active (by default) in our Create View window, and we will not change that.
   
   
2. We will not change the default domain (horizontal) and range (vertical) bounds, but any of them can be changed by:
   1. clicking on the box of the bound to be changed; then
   2. editing the number in the box; and
   3. repeating the steps for any other bounds to be changed.

To create a graph

1. Create the specified view by clicking the Create button, or simply pressing return as the Create button is the default button. (The thick border around the Create button denotes that it is the default button.)
   The graph view and an accompanying view buddy floating window appear as in figure I.7 following:
   
   Figure I.7: Graph view of y=sqrt(x)sin1/x, and the view buddy
   

To apply or remove axes and scales
Figure I.7 shows axes and scales applied to the graph; the steps required to remove or apply them are as follows:

1. Go to the view buddy, click and hold down the mouse on the mode pop-up menu on the top of the floating window; a pull down list appears.
   
   
2. Drag the mouse down the list to Ticks, then release the mouse to enter Ticks mode.
   
   
3. Click on each of the four buttons in turn to see the different ticks. Figure I.7 uses the second button.
   
   
4. The Show Ticks checkbox on top of the buttons is selected. To hide the ticks, click on the checkbox once to toggle ticks inactive. (One more click and the ticks will be toggled active, into the last state selected.)
   
   

To alter the viewport region by zooming
The graph presents a good overview of the relation over the domain and range of [-10, 10]. But the interesting portion seems to be a smaller region around the origin. Although we could have created a view with narrower bounds in the Create View window, it is now more convenient for us to change the viewport region with GrafEq’s zoom feature.

1. Return to the view buddy, and enter zoom mode by selecting Zoom from the mode pop-up menu at the top of the floating window.
   
   
2. The Zoom-Out button is on the view buddy now, but there is not a button for zooming in, since we have to specify the region to be zoomed into.
   
   
3. Place the mouse cursor within the graph view; a zoom box will appear.
   
   
4. Place the zoom box at the graph’s interesting portion, close to the origin, as seen in figure I.8 following:
   
   Figure I.8: Zooming into a graph's interesting area, with the zoom buddy
   

5. Click once to see the selected region, as View #2.
   
   
6. Zoom in to the center once again, to create View #3, as seen in figure I.9 following:
   
   Figure I.9: A good view of the graph's interesting area
   

To edit a relation
We will edit the relation, from y=sqrt(x)sin1/x to y=x²sinx².

1. Bring the relation window back up front, to be the active window: If some part of the window is visible, click on it once. Alternatively, go to GrafEq’s top Graph menu, drag the mouse down to Relation #1(Algebraic), and release the mouse to select the algebraic window for Relation #1.
   

   Hint: The relation window appears, but not the prompt message or the easy button floating window. But the Active box in the relation window is selected, and the graph of Relation #1 is shown in the View #3 window.

2. Click once on the relation, to invoke its edit mode.
   

   Hint: The prompt message appears in the relation window and the easy button floating window appears next to the relation window. But the Active box in the relation window is cleared, and the graph of Relation #1 is not shown in the View #3 window.

3. Place the cursor just in front of “sin”, as seen in figure I.10 following:
   
   Figure I.10: Relation #1 being edited
   

   Tip: Move the cursor by clicking the mouse in the area, or by repeatedly pressing the left arrow key.

   Hint: The relation portion behind the cursor becomes unformatted. Don’t worry about it; it will be automatically reformatted as they pass to the left side of the cursor.

4. Erase the sqrt(x) term by either:
   • highlight the term with the mouse and cursor, then press the del key, or
   • press the delete key until the term is removed.
   
   
5. Enter x² as follows:
   1. Type in x from the keyboard;
   2. Go to the easy button floating window, and click the arrow adjacent to the algebra easy button set to open it;
   3. Click the a² button once, and the relation window will look like figure I.11 following:
      
      Figure I.11: Relation #1 edited
      
   
   
6. Move the cursor to the rightmost end of the relation, and see that the entire relation becomes formatted again, by:
   • clicking the mouse there, or
   • repeatedly pressing the right arrow key until the cursor is there, or
   • holding down the control key and pressing the right arrow key once.
   
   
7. Press the delete key three times to remove the 1/x term.
   
   
8. Enter x² one more time to complete relation editing, as seen in figure I.12 following:
   
   Figure I.12: Relation #1 edited
   

9. Press return to see the new graph plotted in the View #3 window.
   
   Figure I.13: Graph of the edited Relation #1
   

10. The new graph above does not look very interesting. But note that the scales show very small numbers, and remember that we had zoomed in twice to produce View #3; we will now zoom back out to take a look at the “bigger picture”.
    
    
11. Return to the zoom buddy, referring to the steps described earlier when altering the viewport region, press the Zoom Out button a few times. The result of zooming out is intriguing. The following picture, View #7, is obtained after zooming out four times.
    
    Figure I.14: Graph of the edited Relation #1 after four zoom-outs
    

To use constraints in defining a relation
Note that the graph in figure I.14 is much simpler where x is small (less than three). We will now produce a simple graph whose domain is [0,3]. We have learned that:

• we can define the bounds in the Create View window, and that
• we can use the zoom feature of GrafEq.

1. Bring the relation window back to the foremost and invoke its edit mode (refer to the steps described earlier when editing a relation).
   
   
2. The prompt message at the bottom of the window states “....press tab to add a constraint.”. Press the tab key now.
   
   
3. Enter the constraint “0<x<3”, as seen in figure I.15 following:
   
   Figure I.15: Relation #1 with second constraint
   

4. Press return for the resulting graph, as seen in figure I.16 following:
   
   Figure I.16: Graph of Relation #1 with two constraints
   

1. press the mouse button on the File menu, then
2. drag the mouse down the pull-down menu to Close Graph, and release the button to close your graph.

In this section, we will try out graphing multiple relations on the same graph, and a few other features of GrafEq.

To plot simultaneous systems

1. Under the File menu select New Graph, to get a new relation window.
   
   
2. Enter Relation #1 as seen in figure I.17 following, (with the help of the easy buttons for square operations):
   
   Figure I.17: Relation #1 for the simultaneous system
   

3. Press return after Relation #1 is entered correctly, to get to the Create View Window.
   
   
4. The default settings are fine, so press return or click on Create to create a graph.
   
   
5. Under the Graph menu, select New Relation, to get another relation window (entitled Relation #2).
   
   
6. Enter the second relation, referring to figure I.18 following, then press return.
   
   Figure I.18: Relation #2 for the simultaneous system
   

7. Open another relation window (entitled Relation #3), by selecting New Relation under the Graph menu.
   
   
8. According to figure I.19 following, enter the first constraint of the relation, press Tab, then enter the second constraint of the relation:
   
   Figure I.19: Relation #3 for the simultaneous system
   

9. Press return; your graph now shows all three relations in three different colours, layered with Relation #1 at the bottom, then Relation #2, and finally Relation #3 on top, as seen in figure I.20 following:
   
   Figure I.20: Graph of the simultaneous system
   

To blend graphs of simultaneous systems

1. Return to the view buddy, and return to colour mode by selecting Colour from the mode pop-up menu on the top of the floating window.
   
   
2. Click to select the Blend checkbox. The graphs for the simultaneous systems become blended, revealing individual graphs and intersections of the simultaneous system, as seen in figure I.21 following:
   
   Figure I.21: Blended graph of the simultaneous system
   

To determine a point’s coordinates

1. Return to the view buddy, and enter One Point mode by selecting One Point from the mode pop-up menu on the top of the floating window.
   
   
2. Place the mouse close to an intersection point on the graph; pin down the point with the help of the magnified view on the view buddy, as seen following:
   
   Figure I.22: Pin down an intersection point with the one point view buddy
   

3. Read the coordinates of the point from the view buddy.

To change the colours of the graphs

1. Access the colour mode view buddy again.
   
   
2. Click on the colour box of Relation #1. When the colour panel appears, drag the highlight box to choose another colour.
3. Repeat the process to choose colours for any other relations.

To determine distances between points

1. Access the Two Point mode of the view buddy. A line segment AB appears on the graph.
   
   
2. Use the mouse to drag points A and B to the two ends of the distance to be measured. See figure I.23 following as an example:
   
   Figure I.23: Measure distance between two points with the two point view buddy
   

3. Read the distances from the view buddy.

Figure I.24: Reward Relation

Many other features remain to be explored, such as:

• copying and pasting relations and graphs,
• making pages for printing,
• saving graphs to disk,
• using custom ticks,
• using one-key short-cuts, and
• customizing GrafEq for subsequent sessions.

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