9. determine an equation for this cosine function. (3 marks)

Explanation:

Step1: Determine the amplitude

The amplitude $A$ is half the distance between the maximum and minimum values. The maximum value is $4$ and the minimum is $- 4$, so $A=\frac{4 - (-4)}{2}=4$.

Step2: Determine the vertical shift

The mid - line of the function is $y = 0$. So the vertical shift $D = 0$.

Step3: Determine the period

The period $T$ is the distance between two consecutive peaks. Here, $T=\pi$. Since $T=\frac{2\pi}{B}$, then $\pi=\frac{2\pi}{B}$, and solving for $B$ gives $B = 2$.

Step4: Determine the phase shift

The cosine function starts at its maximum value at $x = 0$ (no phase shift), so $C = 0$. The general form of a cosine function is $y=A\cos(Bx - C)+D$. Substituting $A = 4$, $B = 2$, $C = 0$ and $D = 0$ into the formula, we get $y = 4\cos(2x)$.

Answer:

$y = 4\cos(2x)$