### 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
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Examples
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Other Notes

**How to use**

- Enter a function
*f*_{1}(*x*) in the text input field
marked "*f*_{1}(*x*)="
(Example:
*f*_{1}(*x*)=*x*^{2})
- 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 *f*_{1}(*x*)

- To see two functions graphed simultaneously:
- Enter the second function
*f*_{2}(*x*) in the text input field
marked "*f*_{2}(*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
*f*_{2}(*x*) -- changing the value of *c*
affects the green points on
the graphs of *both* functions simultaneously

(Example:
*f*_{1}(*x*)=*x*^{2}
and *f*_{2}(*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 *f*_{1}(*x*) and
*f*_{2}(*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**

**Other Notes**

The labels marked "*f*_{1}(*c*)=" and
"*f*_{2}(*c*)=" give the values of
the functions *f*_{1}(*x*) and
*f*_{2}(*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, 3*pi*/4, 5*pi*/4, etc.