First question

So you have

Do you know about derivatives? Taking the derivative will help.

First expand:

Take the first derivative:

This is a new function describing the original function's slope given an . The vertex of the parabola will have a slope of zero, so set and solve for . This will yield the x-value of the vertex, and you can plug it back into the original function to find its corresponding y.

Take the dervative one more time:

This new function gives you the the rate with which the slope of the original function changes. It's useful to determine concavity in a situation like this. As a rule of thumb, if the function is a positive constant, then the parabola opens upward. If it's negative, then it opens downward. In this case, it's positive.