This applet is designed to allow a visual exploration of a family of function, or how shifting a parameter changes the graph.[br][br]The applet opens with the function[br] [math]f(x)=a*x^2+b\cdot x+c[/math]. [br]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. [br][br]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.[br][br][list][*]The first examples graph quadratic functions. They can be expressed in a number of ways, each of which has advantages:[list][*]The general format -- f(x)=a*x^2+b*x+c [br]A nice polynomial format, however the geometric understanding of the parameters is harder.[br][/*][*]Vertex format -- f(x) = a*(x-b)^2+c [br]The parameters give the vertex and direction.[br][/*][*]Intercept format -- f(x) = a*(x-b)*(x-c) [br]The parameters give the x intercepts.[/*][/list][/*][*]There are several examples that are related to trig functions.[list][*]sin curve -- f(x)=a sin(b(x-c)) [br]The parameters are related to amplitude, period and shift.[br][/*][*]Linear combinations of sin(x) and cos(x) -- f(x) = a*sin(x) + b*cos(x)[br]This looks like a sin curve with a shift based on the ration of a and b.[br][/*][*]Combinations with different periods - In the user defined functions, it is worthwhile to look at[br]f(x)=a*sin(x)+b*sin(c*x)[/*][/list][/*][*]Logrithmic curves -- f(x) = a*ln(b*x)+c[br]Pay attention to the signs of a and b and the value of c[br][/*][*]Exponential curve -- f(x)=a*b^x+c[br][/*][*]Cubic curve -- f(x)=(x-a)(x-b)(x-c)[br][/*][*]Rational function curve -- f(x)=((x-a)(x-b))/(x^2-c^2)[br]Notice how the shape chances if a and b are inside or outside of the interval from -c to c[br][/*][*]User choce - any function with up to three parameters.[br]Rational functions -- f(x) = (x*(x-a))/((x-b)*(x-c))[br][/*][/list]

Enter a function of [math]x [/math].[br]Choose the degree of the polynomial by sliding point [math]n[/math] on the slide bar.[br]Choose the center of the polynomial by sliding point [math]a [/math] on the slide bar.

This applet is designed to illustrate the shell method for solids of revolution.[br]There are three windows:[br][br]The first window shows the diagram in the x-y plane.  There is an upper and lower function.[br]Draggable points let you control the limits of integration, the axis of revolution, and the position of the line that will become the sample shell.[br][br]The second window is the 3-D window that shows the results of the rotation.[br][br]The third window has controls and results.  has a place for entering the upper and lower functions.  It gives the area of the current shell, both generically and for the current location.  It gives and evaluates the integral for volume, both generically and as a number, both up to the current shell, and for the whole volume.[br]Controls let you determine how much of the solid is shown and the viewing window.

Some more text below.

[list=1][br][*]Enter different integers (whole numbers).[br][*]See what prime numbers compose your integer.[br][*]Press the prime factors buttons and see how your number can be produced by multiplying different pairs of numbers.[br][*]How many different pairs that produce your number are there?[br][/list]