Problem

Produce several ( 5 to 10) graphs of on
the same axes using different values for **d**. Does varying
**d** change the shape of the graph? the position?

Note that varying *d *just translates
each function along the x-axis. Varying *d* is a horizontal
translation on the graph. Observe that there is not change in
the shape, there only is a change in the position of the graphic.

Now again consider the equation *. *It
follows that the derivative is

There are some important details here.

First, for all values of *d *indicated
in the graph above the zeros for the corresponding derivatives
all pass through the line of symmetry.

Secondly, we can find an extreme value for
. So, for *y' = 2(x-d)* *, *its zero is *x = d*.
If we substitute in the original equation we would get y=-2. This
means that an extreme value of the parabola is in y=-2.

We can see that the parabola moves through y=-2