### Tracing Graphs of Functions

Graphs up to two functions simultaneously, and traces the graphs of the functions, to explore values of functions and points of intersection.
How to use   ||   Examples   ||   Other Notes

How to use
• Enter a function f1(x) in the text input field marked "f1(x)=" (Example: f1(x)=x2)
• Click the "Graph" button (this button also refreshes the graph)
• Functions can be traced either of two ways:
• Enter a value into the text input field marked "c=" and click the "Graph" button
• The value of c can also be specified by using the mouse to click and drag the green point on the graph -- this point takes its horizontal position from the mouse position, and follows the graph of f1(x)
• To see two functions graphed simultaneously:
• Enter the second function f2(x) in the text input field marked "f2(x)="
• Click (to its "on" state) the check box next to this input field
• Click the "Graph" button
• A green point appears also on the graph of f2(x) -- changing the value of c affects the green points on the graphs of both functions simultaneously
(Example: f1(x)=x2 and f2(x)=2-x)
Up to 2 functions can be graphed and traced simultaneously
• To remove a function from the graph, click (to its "off" state) the check box next to the associated text input field and click the "Graph" button to refresh
• To erase the graph and all input fields, click the "Clear" button
The text input fields for f1(x) and f2(x) can accept a wide variety of expressions to represent functions, and the buttons under the graph allow various manipulations of the graph coordinates.
Examples
 Intersecting Lines:    f1(x)=x-1 f2(x)=2-x/2 Line and Parabola:    f1(x)=x2 f2(x)=2-x) Sine and Cosine:    f1(x)=sin x f2(x)=cos x)

Other Notes
The labels marked "f1(c)=" and "f2(c)=" give the values of the functions f1(x) and f2(x) when x=c, where c is the value given in the field marked "c=" (the horizontal position of the green points on the graph).

By using the tracing actions with the green points on the graphs, it is possible to approximate and explore intersection points of graphs. To find such points more precisely, combine the tracing actions with using the "Zoom In" button under the graph. All of the examples above have graphs which intersect. In the "Sine and Cosine" example above, the graphs intersect at points corresponding to values c=pi/4, 3pi/4, 5pi/4, etc.