Graphing On The Geometer's Sketchpad

1. Open the Graph menu and click on Create Axes.
2. Click on the x-axis. Open the  Construct Menu and  click on  Point on Object.This will create an arbitrary point, which we label as C.
3. Click on the point C and also on the x-axis. (Remember to hold the shift key down when highlighting more than one object.)
4. Open the Edit Menu followed by the Action Button line and then the line labelled Animation. A menu called Path Match will appear. By default the menu will read:  Point C moves bidirectionally along the axis x quickly.
5. Open the direction icon and change the direction that C moves from bidirectionally to one way}.  Click on Animate.
6. Click on C. Open the Measure menu and click on Coordinates.. Highlight the coordinates of C, open the Measure menu and click on Calculator. Open the values icon, click on the point C and drag to the x-coordinate of C. The x-coordinate of C, which is labelled as x_C, will appear on your screen.
7. Relabelling x_C: The label x_C looks awkward. We can relabel x_C to x as follows. Highlight x_C. Give x_C the finger! This means double click on x_C with the text tool. The Format Measurement menu appears. Click on Text Format. Click on the box with x[C] = . Type over this with ``x space = space". You must put a single typewriter space before the equal sign and after.
8. You are now ready to begin graphing. As an illustration, we will  graph the cubic polynomial                                                       f(x) = x(x - 1)(x - 2)  .
9. Load the  x-coordinate into the calculator and perform the multiplication f(x). When finished, click on OK and the value f(x) will appear on your screen. The value on yourscreen will read ``x(x - 1)(x - 2) =".  Following the pattern in step 5, you may want to relabel this computation as ``f(x)".
10. Highlight x and then the computation f(x) from step 6. Open the Graph Menu} and click on Plot As (x,y). A new point will appear on your graph. This is the point (x, f(x)).
11. Highlight the plotted point from step 6, open the Display Menu and relabel this point as (x, f(x)). Highlight the point (x, f(x)) again, open the Display Menu} and click on the line Trace Point. If you want, you can color the point (x,f(x)) so that when you run the animation, the graph of the function will be colored. So highlight (x, f(x)), open the display menu and then the color menu.
12. Double click on your animation and watch the graph evolve. Clicking will stop the animation and the graph will appear.
13. The (x, f(x))-rectangle: Construct the segment with endpoints C and (x, f(x)). Construct the line l that is parallel to the x-axis and passing through the point (x, f(x)). Construct the intersection point E of l and the y-axis. Hide the line l and then construct the segment with endpoints E and (x, f(x)). Run the animation again. Watch the rectangle change.

Challenge: In order to understand the geometry of the derivative of a function, we approximate the tangent lines to the graph of f with the idea of secant lines. The challenge is to construct secant lines on the graph of f?