Understanding Quadratic Functions

A quadratic function
f(x) = ax² + bx + c
is specified by the three constants a,b, and c, which we will refer to as parameters. By completing the square, we can express this same function by
f(x) = a(x-h)² + k
so that the quadratic function f(x) is equally well specified by the three parameters a, h, k. It's easier, though, to understand how a,h, k are related to the parabola that is the graph of f(x).

The applet below will draw the graph of   f(x) = a(x-h)² + k   for values of a,h,k that you can select (between -5 and 5) by moving the sliders or clicking  the arrows at the ends of the sliders.

   Describe the effect of changing each of a, h, and k on the shape or position of the parabola. Be as specific as you can.

   Can you adjust the parameters so the resulting parabola goes through the centers of the three crosses? Do you think that this is possible for any three crosses whose centers have different x-coordinates?

Now let's go back to the original expression for f(x)  and write it in its original form:

f(x) = a² + bx +c

Use the applet below to answer these questions:

How does changing a,b,c affect the parabola?

Describe how changing b affects the shape or position of the parabola. Be as specific as you can. Try adjusting the parameters to make the graph pass through the 3 cross centers.

Is this easier or harder than with the applet above which uses a,h, and k as parameters?