Both F(x) and G(x) are quadratic equations. The graphs of the two functions are known as parabolas. All four choices for the equation of F(x) are written in their vertex form. That is:
The point is the vertex of the parabola, and
The value of determines the width and the direction of the opening of the parabola. means that the parabola opens upward. means that the parabola opens downwards. The opening becomes narrower if the value of increases.
It's a method called "completing the square". We add/subtract an appropriate constant term that allows us to condense three terms into a squared binomial, and leaves a constant remainder outside the squared term.
In this case, we first factor out -16 from the first two terms:
When we square a binomial like , we end up with the expansion , so that in the quadratic above we need with and . Then the constant term we have to add/subtract will be :
Now the first three terms within the parentheses can be condensed to get
and all that's left to do is distribute a factor of -16 and combine the remaining constant term:
Then the vertex of the parabola defined by this expression occurs at .