8.
$$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$
- Write down the maximum value of \(\mathrm { f } ( x )\).
- Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points at which the graph meets the coordinate axes.
The points \(A\) and \(B\) on the graph of \(y = \mathrm { f } ( x )\) have coordinates \(( - 2 , - 7 )\) and \(( 3,8 )\) respectively.
- Find, in the form \(y = m x + c\), an equation of the straight line through \(A\) and \(B\).
- Find the coordinates of the point at which the line \(A B\) crosses the \(x\)-axis.
The mid-point of \(A B\) lies on the line with equation \(y = k x\), where \(k\) is a constant.
- Find the value of \(k\).