sketch a graph of the function $f(x)= - 5sin(\frac{pi}{3}x)$

Explanation:

Step1: Identify the amplitude

The general form of a sine - function is $y = A\sin(Bx - C)+D$. For the function $f(x)=- 5\sin(\frac{\pi}{3}x)$, the amplitude $|A| = |-5|=5$. This means the graph will oscillate between $y = 5$ and $y=-5$.

Step2: Find the period

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

Step3: Determine key points

• When $x = 0$, $f(0)=-5\sin(0)=0$.
• When $x=\frac{3}{2}$, $f(\frac{3}{2})=-5\sin(\frac{\pi}{3}\times\frac{3}{2})=-5\sin(\frac{\pi}{2})=- 5$.
• When $x = 3$, $f(3)=-5\sin(\pi)=0$.
• When $x=\frac{9}{2}$, $f(\frac{9}{2})=-5\sin(\frac{3\pi}{2}) = 5$.
• When $x = 6$, $f(6)=-5\sin(2\pi)=0$.

Step4: Sketch the graph

Plot the key points $(0,0),(\frac{3}{2},-5),(3,0),(\frac{9}{2},5),(6,0)$ and connect them with a smooth sine - wave curve. The negative sign in front of the 5 reflects the graph of $y = 5\sin(\frac{\pi}{3}x)$ about the $x$ - axis.

Answer:

Sketch the graph using the key - points $(0,0),(\frac{3}{2},-5),(3,0),(\frac{9}{2},5),(6,0)$ and a smooth sine - wave curve with amplitude 5 and period 6, reflected about the $x$ - axis.