7

7. [Maximum mark: 16]
   The following diagram shows parts of the graphs of two functions $f$ and $g$.

The graph of $f$ is linear, has an x‑intercept at $(4,0)$ and a y‑intercept at $(0,8)$.

(a) Write down the equation for $f$ in the form $f(x)=mx+c$. [2]

The function $g$ is given by $g(x) = -x^{2} + bx + 8$, where $b$ is a real constant.

(b) Find the value of $b$. [3]

(c) Show that the area of the region enclosed by the graph of $f$ and the graph of $g$ can be represented by the definite integral $\displaystyle\int_{0}^{4} (-x^{2}+4x)\,dx$. [2]

(d) Hence, find the area of the region enclosed by the graph of $f$ and the graph of $g$. [4]

Point $P$ is on the graph of $g$. The tangent to the graph of $g$ at $P$ is parallel to the graph of $f$.

(e) Find the coordinates of $P$. [5]