Question

the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer.
which of the following is the graph of ( y = - 3 cdot sin left( \frac { 1 } { 3 } x
ight) )?

Explanation:

Step1: Recall the general form of a sine function

The general form of a sine function is \(y = A\sin(Bx - C)+D\). For the function \(y=-3\sin(\frac{1}{3}x)\), we have \(A=-3\), \(B = \frac{1}{3}\), \(C = 0\), \(D=0\).

Step2: Determine the amplitude

The amplitude is \(|A|\). Here, \(|A|=|-3| = 3\).

Step3: Determine the period

The period of a sine function \(y = A\sin(Bx)\) is \(T=\frac{2\pi}{|B|}\). Substituting \(B=\frac{1}{3}\), we get \(T=\frac{2\pi}{\frac{1}{3}}=6\pi\).

Step4: Analyze the sign of \(A\)

Since \(A=-3<0\), the graph of \(y = \sin(x)\) is reflected about the \(x -\)axis.

Answer:

The second graph (the one that has a period of \(6\pi\), amplitude \(3\) and passes through the origin \((0,0)\) with the negative - coefficient of the sine function causing the appropriate reflection) is the graph of \(y=-3\sin(\frac{1}{3}x)\).