Question

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.

Answer 1

$y=Asin\left(B\theta\right)$

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:

$y=\frac{1}{2}sin\left(3\theta\right)$

Amplitude:

$A=\frac{1}{2}$

Period:

$P=\frac{2\pi}{3}$

Points of intersection with the x-axis:

$\begin{gathered} \frac{1}{2}sin\left(3\theta\right)=0 \\ \\ Multiply\text{ both sides of the equatio by 2:} \\ 2*\frac{1}{2}sin\left(3\theta\right)=2*0 \\ \\ sin\left(3\theta\right)=0 \end{gathered}$

Using the unit circle you get that the angles with sine equal to 0 are: 0 and π.

$\begin{gathered} 3\theta=0 \\ 3\theta=\pi \end{gathered}$

Solve θ:

$\begin{gathered} \theta=0 \\ \\ \theta=\frac{\pi}{3} \end{gathered}$

Add the period to each solution multiplied by k of θ to find all the intersections:

$\begin{gathered} \theta=0+\frac{2k\pi}{3} \\ \\ \theta=\frac{\pi}{3}+\frac{2k\pi}{3} \end{gathered}$

Combine the solutions:

$\theta=\frac{k\pi}{3}$

Then, the given function has Amplitude 1/2; period 2π/3, and points of intersection kπ/3 (k is a whole number)