Question

which of the following is the graph of y = - 3·sin(\frac{1}{3}x)?

Explanation:

Step1: Analyze the amplitude

The general form of a sine - function is $y = A\sin(Bx)$. Here, $A=- 3$, so the amplitude $|A| = 3$. This means the maximum value of the function is $y = 3$ and the minimum value is $y=-3$.

Step2: Analyze the period

The period of the sine - function $y=\sin(Bx)$ is given by $T=\frac{2\pi}{|B|}$. For the function $y=-3\sin(\frac{1}{3}x)$, $B = \frac{1}{3}$, so the period $T=\frac{2\pi}{\frac{1}{3}}=6\pi$.

Step3: Analyze the reflection

Since $A=-3$, the graph of $y = \sin(\frac{1}{3}x)$ is reflected about the $x$ - axis.

The graph that has an amplitude of 3, a period of $6\pi$ and is reflected about the $x$ - axis is the correct one.

Answer:

We need to visually identify the graph among the options that has an amplitude of 3, a period of $6\pi$ and is reflected about the $x$ - axis. Without the actual labels on the options, we can't specify which one it is exactly, but the correct graph should have these characteristics: it oscillates between $y = 3$ and $y=-3$, has a period of $6\pi$ (the distance between two consecutive peaks or troughs is $6\pi$) and is upside - down compared to the standard $y=\sin(x)$ graph.