Question

θ | f(θ)
0 | 0
\frac{3π}{2} | 4
3π | 0
\frac{9π}{2} | -4
6π | 0
what equation would the table of values help you graph?
(1 point)
\bigcirc y = 4\sin\left(\frac{θ}{3}\
ight)
\bigcirc y = 6\sin\left(\frac{3θ}{2}\
ight)
\bigcirc y = 4\cos\left(\frac{θ}{3}\
ight)
\bigcirc y = 4\sin(θ)

Explanation:

Step1: Analyze the general form of sine function

The general form of a sine function is \(y = A\sin(B\theta)\), where \(A\) is the amplitude (\(\vert A\vert\) is the maximum value of the function) and the period \(T=\frac{2\pi}{B}\).
From the table, the maximum value of \(y = f(\theta)\) is \(4\), so \(A = 4\).

Step2: Calculate the period

The period \(T\) is the distance between two consecutive same - value points (e.g., from \(\theta = 0\) to \(\theta=6\pi\), the function repeats its values). So \(T = 6\pi\).
Using the formula \(T=\frac{2\pi}{B}\), we substitute \(T = 6\pi\) into it:

$$6\pi=\frac{2\pi}{B}$$

Solve for \(B\):

$$B=\frac{2\pi}{6\pi}=\frac{1}{3}$$

Answer:

\(y = 4\sin(\frac{\theta}{3})\)