a) notice that the curve is symmetrical about the line x = 5.

Since P and Q have the same y coordinate, they are a reflection of each other. Therefore they are equidistant from x = 5.

So, the x coordinate of the midpoint of P and Q is 5.

(2 + q)/2 = 5
2 + q = 10
q = 8


OR


Horizontal Distance from line x = 5 to P
= 5 units - 2 units = 3 units.

Since
Horizontal Distance from line x = 5 to P =
Horizontal Distance from line x = 5 to Q,

q = 5 + 3 = 8

Quadratic curve can be represented by the formula y = p(x - q)² + r, where (q,r) are the coordinates of the turning point.

Since y = ∣ a(x + b)² + c ∣, if you compare the above equation,

Realise that b = - q and c = r

The turning point has coordinates (5,8). But as the curve = ∣a(x + b)² + c∣ ,
(5,8) is the reflection of the negative portion of the curve below the y axis.

So in the curve of y = a(x + b)² + c,
the turning point would be (5,-8)

The equation of the curve can be rewritten as y = ∣ a(x - 5)² - 8 ∣

So c = -8, b = -5

For a, substitute the coordinates of P or Q to find it.


c) and d) are easy. Draw some lines to see where the condition can be satisfied (visualising also works)