question 1 (5 points)\nsketch a graph of a periodic function with the following properties (make sure your…

Explanation:

Step1: Recall sine - like function form

A general form of a periodic function is $y = A\sin\left(\frac{2\pi}{T}x\right)$ (we can also use cosine - like form), where $A$ is the amplitude and $T$ is the period. Given $A = 4$ and $T=6$. The function becomes $y = 4\sin\left(\frac{2\pi}{6}x\right)=4\sin\left(\frac{\pi}{3}x\right)$.

Step2: Determine key points

For $y = 4\sin\left(\frac{\pi}{3}x\right)$:

• When $x = 0$, $y=4\sin(0)=0$.
• When $x=\frac{3}{2}$, $y = 4\sin\left(\frac{\pi}{3}\times\frac{3}{2}\right)=4\sin\left(\frac{\pi}{2}\right)=4$.
• When $x = 3$, $y=4\sin(\pi)=0$.
• When $x=\frac{9}{2}$, $y = 4\sin\left(\frac{\pi}{3}\times\frac{9}{2}\right)=4\sin\left(\frac{3\pi}{2}\right)= - 4$.
• When $x = 6$, $y=4\sin(2\pi)=0$.

Step3: Sketch the graph

Start at the origin $(0,0)$. Then go up to the point $\left(\frac{3}{2},4\right)$, come back down to $(3,0)$, go down to $\left(\frac{9}{2}, - 4\right)$ and then back up to $(6,0)$. Repeat this pattern for other intervals since the period is 6. Mark the $x$ - axis with values at intervals of 1 unit (or other appropriate scale) and the $y$ - axis with values from - 4 to 4. Label the amplitude (maximum and minimum values) and the period (show one full cycle from $x = 0$ to $x = 6$).

Answer:

A hand - drawn graph of $y = 4\sin\left(\frac{\pi}{3}x\right)$ with key points labeled as described above, showing one full cycle from $x = 0$ to $x = 6$ and amplitude of 4.