The quadratic function forms a parabola. The vertex form of the equation is expressed as
y = a(x - h)^2 + k
Where
h and k are the x and y coordinates of the parabola's vertex. Given that the vertex is (4, 5),
h = 4, k = 5
Substituting these values into the above equation, it becomes
y = a(x - 4)^2 + 5
Given that the parabola passes through the point, (1, 2), we would substitute x = 1 and y = 2 into y = a(x - 4)^2 + 5. It becomes
2 = a(1 - 4)^2 + 5
2 = a * 9 + 5
2 = 9a + 5
9a = 2 -5
9a = - 3
a = - 3/9 = - 1/3
Substituting a = - 1/3 into y = a(x - 4)^2 + 5, the equation would be
[tex]y\text{ = -}\frac{1}{3}(x-4)^2\text{ + 5}[/tex]