A wave is modeled with the function y = 1/2 sin (3Θ) , where Θ is in radians. Describe the graph of this function, including its period, amplitude, and points of intersection with the x-axis.

In the function above:
A is the amplitude
2π/B is the period
You equal to 0 the function and solve θ to find the points of intersection with the x-axis.
For the given function:
[tex]y=\frac{1}{2}sin\left(3\theta\right)[/tex]Amplitude:
[tex]A=\frac{1}{2}[/tex]Period:
[tex]P=\frac{2\pi}{3}[/tex]Points of intersection with the x-axis:
[tex]\begin{gathered} \frac{1}{2}sin\left(3\theta\right)=0 \\ \\ Multiply\text{ both sides of the equatio by 2:} \\ 2*\frac{1}{2}sin\left(3\theta\right)=2*0 \\ \\ sin\left(3\theta\right)=0 \end{gathered}[/tex]Using the unit circle you get that the angles with sine equal to 0 are: 0 and π.
[tex]\begin{gathered} 3\theta=0 \\ 3\theta=\pi \end{gathered}[/tex]Solve θ:
[tex]\begin{gathered} \theta=0 \\ \\ \theta=\frac{\pi}{3} \end{gathered}[/tex]Add the period to each solution multiplied by k of θ to find all the intersections:
[tex]\begin{gathered} \theta=0+\frac{2k\pi}{3} \\ \\ \theta=\frac{\pi}{3}+\frac{2k\pi}{3} \end{gathered}[/tex]Combine the solutions:
[tex]\theta=\frac{k\pi}{3}[/tex]Then, the given function has Amplitude 1/2; period 2π/3, and points of intersection kπ/3 (k is a whole number)