2023 ACS (BR) 3AM EOY Q4

An arched underpass has the shape of a parabola as shown. In the diagram, the curved underpass can be modelled by the quadratic function \({y = a(x - h)^2 + k}\), where \({a}\), \({h}\), and \({k}\) are constants. In this model, \({x}\) m is the horizontal distance from \({O}\) and \({y}\) m is the height of the arch.

1. A one-way road passing through the arch is \({4.2}\) m wide and the maximum height of the arch is \({4.2}\) m. State the coordinates of the maximum point of the quadratic function.[1]
2. Hence find the exact value of \({a}\).[2]
3. Determine whether it is possible for a vehicle that is \({2.8}\) m wide and \({2.4}\) m tall to navigate through the underpass. Explain your reasoning by showing your workings clearly.[3]