i)
P is the vertex of the graph, which means it is the highest point of the graph, whereby the value of y is the maximum at P.
Since the question says the maximum value of the function is 9, y = 9 at P. P is what we call the maximum point.
y = -(x - 4)² + k
For the maximum point, we want the max value of y. This means that the negative value in the equation above, -(x-4)² should be as small as possible. k is a constant so we don't have to worry about it.
This is achieved when x = 4, which turn means that y = -(4-4)² + k = -0² + k = k
At other values of x, you will always get a negative value for -(x-4)², because (x - 4)² itself will be always positive (the square of any real non-zero value is always > 0) and the negative sign outside makes it negative.
Eg. When x = 0, y = -(0 - 4)² + k = -(-4)² + k = -16 + k = k - 16
When x = 7, y = -(7 - 4)² + k = -(3)² + k = -9 + k = k - 9
When x = -2, y = -(-2 - 4)² + k = -(-6)² + k = -36 + k = k - 36
These are all smaller than k (when x = 4)
Therefore, since y = 9, x = 4 at the maximum point,
9 = -(4-4)² + k
9 = 0 + k
k = 9

y = a(x-h)² + k is known as the vertex form of a parabola, whereby:
①a is the coefficient of x² (negative means the graph is downward sloping, n-shaped and positive means the graph is upward sloping, u-shaped)
② (h,k) is the coordinates of the vertex. In this question, h = 4, k = 9, a = -1.

For further reading :
https://www.purplemath.com/modules/sqrvertx.htm