Question

complete the table for f(θ) for the given values of θ

θ

f(θ)

|0|
|$\frac{pi}{4}$|
|$\frac{pi}{2}$|
|$\frac{3pi}{4}$|
|$pi$|
use the table of values to graph the f(θ)=4 sin(2θ) and find the amplitude, midline, and period. (1 point)
amplitude=2, midline: y = 0, period=$\frac{pi}{4}$
amplitude=2, midline: y = 4, period=$\frac{pi}{2}$
amplitude=4, midline: y = 0, period=$pi$
amplitude=4, midline: y = 2, period=$\frac{pi}{4}$

Explanation:

Step1: Calculate \(f(\theta)\) for \(\theta = 0\)

Substitute \(\theta=0\) into \(f(\theta)=4\sin(2\theta)\), we get \(f(0)=4\sin(2\times0)=4\sin(0) = 0\).

Step2: Calculate \(f(\theta)\) for \(\theta=\frac{\pi}{4}\)

Substitute \(\theta = \frac{\pi}{4}\) into \(f(\theta)=4\sin(2\theta)\), then \(2\theta=2\times\frac{\pi}{4}=\frac{\pi}{2}\), and \(f(\frac{\pi}{4})=4\sin(\frac{\pi}{2})=4\times1 = 4\).

Step3: Calculate \(f(\theta)\) for \(\theta=\frac{\pi}{2}\)

Substitute \(\theta=\frac{\pi}{2}\) into \(f(\theta)=4\sin(2\theta)\), so \(2\theta = 2\times\frac{\pi}{2}=\pi\), and \(f(\frac{\pi}{2})=4\sin(\pi)=4\times0 = 0\).

Step4: Calculate \(f(\theta)\) for \(\theta=\frac{3\pi}{4}\)

Substitute \(\theta=\frac{3\pi}{4}\) into \(f(\theta)=4\sin(2\theta)\), then \(2\theta=2\times\frac{3\pi}{4}=\frac{3\pi}{2}\), and \(f(\frac{3\pi}{4})=4\sin(\frac{3\pi}{2})=4\times(- 1)=-4\).

Step5: Calculate \(f(\theta)\) for \(\theta=\pi\)

Substitute \(\theta=\pi\) into \(f(\theta)=4\sin(2\theta)\), so \(2\theta=2\pi\), and \(f(\pi)=4\sin(2\pi)=4\times0 = 0\).

Step6: Find the amplitude, mid - line and period

For a sine function of the form \(y = A\sin(B\theta)+C\), the amplitude is \(|A|\), the mid - line is \(y = C\) and the period is \(T=\frac{2\pi}{|B|}\). For \(f(\theta)=4\sin(2\theta)\), \(A = 4\), \(B = 2\) and \(C = 0\). The amplitude is \(|4|=4\), the mid - line is \(y = 0\) and the period is \(T=\frac{2\pi}{2}=\pi\).

Answer:

\(\theta\)	\(f(\theta)\)
\(\frac{\pi}{4}\)	\(4\)
\(\frac{\pi}{2}\)	\(0\)
\(\frac{3\pi}{4}\)	\(-4\)
\(\pi\)	\(0\)

The correct option for amplitude, mid - line and period is: amplitude = 4, midline: \(y = 0\), period=\(\pi\) (the third option).