Question

4.1 write down the amplitude of g. (1)
4.2 determine the period of f. (1)
4.3 determine the values of a, b and p. (3)
4.4 determine the values of x such that:
4.4.1 g(x)>0 (1)

Explanation:

Step1: Recall amplitude definition

The amplitude of a periodic function is half the difference between the maximum and minimum values. For function $g$, the maximum value is close to $0$ and the minimum value is close to $- 2$. Amplitude $A=\frac{0 - (-2)}{2}$.
$A = 1$

Step2: Recall period definition

The period of a periodic function is the horizontal distance between two consecutive identical points. For function $f$, from one peak to the next peak, it goes from approximately $- 30^{\circ}$ to $150^{\circ}$. Period $T=150^{\circ}-(-30^{\circ})$.
$T = 180^{\circ}$

Step3: Assume general form of function

Assume $f(x)=a\sin(bx + p)$. The amplitude of $f$ is $2$, so $a = 2$. The period $T=\frac{360^{\circ}}{|b|}$, since $T = 180^{\circ}$, then $180^{\circ}=\frac{360^{\circ}}{|b|}$, so $b = 2$. When $x = 0$, $f(0)=2\sin(p)=0$, and from the graph, the phase - shift makes sense when $p = 0$. So $a = 2,b = 2,p = 0$.

Step4: Find positive - value intervals

We look at the graph of $g(x)$ and find the $x$ - values for which the graph is above the $x$ - axis. From the graph, $g(x)>0$ when $120^{\circ}

Answer:

4.1. $1$
4.2. $180^{\circ}$
4.3. $a = 2,b = 2,p = 0$
4.4.1. $120^{\circ}