Quadratic Functions, Equations and Inequalities - Completing the Square and Vertex Form

The cross-section of a skateboard half-pipe is modelled by the quadratic function \(y = \frac{1}{2}x^2 - 4x + 9\), where \(x\) metres represents the horizontal distance from left end of the half-pipe, and \(y\) metres represents the height above the ground.

The graph of the function models the curved surface of the half-pipe.

1. Show that the equation of the curve can be written in the form \(y = \frac{1}{2}(x - 4)^2 + 1.\)

   Hence, state the coordinates of the turning point.

2. Explain what the turning point represents in the context of the half-pipe.
3. Sketch the graph of the quadratic function clearly showing:
   • the turning point,

   • the intercept with the \(y\)-axis,

   • the axis of symmetry.
4. If the right end of the half pipe has a height of \(13.5\) m from the ground, find the width of the half pipe from one end to the other.