What is the equation of the following graph in vertex form? parabolic function going down from the left through the point zero comma twelve and through the point two comma zero and turning at the point four comma negative four and going up through the point six comma zero and continuing towards infinity

Hi there!

The vertex form equation is y=a(x-d)^2+c where the vertex is (d, c).
The vertex is always the turning point when the y-values start to travel in the opposite direction from which they began traveling on the opposite side of the turning point. Therefore the point (4, -4) is the vertex. 
y=a(x-4)^2-4
We can sub in a point on the graph in order to solve for the a-value. Let's use (6, 0), and sub that in for x and y.
0=a(6-4)^2-4
0=a(2)^2-4
4=4a
a=4/4
a=1

Therefore the equation is y=(x-4)^2-4

Hope this helps!