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