This applet is designed to allow a visual exploration of a family of function, or how shifting a parameter changes the graph.
The applet opens with the function f(x) = a*x^2 + b*x + c. The sliders on the side let you change the parameters. The sliders in the bottom window let you change the viewing window, the function, or the base point. The last choice of functions is a user defined function.

As mentioned above, the applet comes preloaded with several families of functions. With each family, it is useful to ask about what features of the graph are noteworthy and how they change with the change in parameters.

The first examples graph quadratic functions. They can be expressed in a number of ways, each of which has advantages:

The general format -- f(x)=a*x^2+b*x+c
A nice polynomial format, however the geometric understanding of the parameters is harder.

Vertex format -- f(x) = a*(x-b)^2+c
The parameters give the vertex and direction.

Intercept format -- f(x) = a*(x-b)*(x-c)
The parameters give the x intercepts.

There are several examples that are related to trig functions.

sin curve -- f(x)=a sin(b(x-c))
The parameters are related to amplitude, period and shift.

Linear combinations of sin(x) and cos(x) -- f(x) = a*sin(x) + b*cos(x)
This looks like a sin curve with a shift based on the ration of a and b.

Combinations with different periods - In the user defined functions, it is worthwhile to look at
f(x)=a*sin(x)+b*sin(c*x)

Logrithmic curves -- f(x) = a*ln(b*x)+c
Pay attention to the signs of a and b and the value of c

Exponential curve -- f(x)=a*b^x+c

Cubic curve -- f(x)=(x-a)(x-b)(x-c)

Rational function curve -- f(x)=((x-a)(x-b))/(x^2-c^2)
Notice how the shape chances if a and b are inside or outside of the interval from -c to c

User choce - any function with up to three parameters.
Rational functions -- f(x) = (x*(x-a))/((x-b)*(x-c))