Parabola

Vertex form    (3,1)

 

y = a ( x -3)^2 +1       point 0,7      shows a =     2/3

 

y = 2/3 (x-3)^2 +1        now expand the right side to find a b and c

We can graph the parabola using vertex form: \(y = a(x-h)+k\)

 

We know the vertex is \((3,1)\). Plugging these values in for h and k, we get: \(y=a(x-3)^2+1\)

 

To solve for a, we have to plug in coordinates. In this case, I will use \((0, 7)\), but you will get the same answer with a different point.

 

We now have: \(7=a(-3)^2+1\)

 

Solving, we find \(a = {2 \over 3}\)

 

Now, we have: \(y = {2 \over 3} (x-3)^2+1\)

 

Expanding the right-hand side, we get: \(y = -{2\over3}x^2-4x+7\)

 

Now, we have: \(-{2 \over 3} - (-4)+7 = \color{brown}\boxed{10 {1 \over3}}\)