Finish the steps below to write a quadratic function for
the parabola shown.
1. Use the vertex form, f(x) = a(x – h)2 + k, and
substitute in the values for h and k.
f(x) = a(x – 5)2 + 3
2. Use another point and substitute in values
for x and f(x). Solve for a.
5 = a(6 – 5)2 + 3
3. Write the function, using the values for h, k,
and a.
The function is f(x) =
12+

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

i may be incorrect but i believe that her first error was in step one when she multiplyed her equation by 1/4 when she should have divided it by 1/4