2023 ACSBR AM S3 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]