3 multiple choice 1 point write the equation for the sine function shown below. y = 4 sin 6θ y = 4 sin 3θ y…

Explanation:

Step1: Determine the amplitude

The amplitude (A) of a sine function (y = A\sin(B\theta)) is the maximum distance from the mid - line. The mid - line is (y = 0). The maximum value is (4) and the minimum value is (-4), so (|A|=4).

Step2: Determine the period

The general formula for the period of (y=\sin(B\theta)) is (T=\frac{2\pi}{B}). From the graph, in the interval (\theta\in[0, 2\pi]), the number of complete cycles (n = 3). The period (T=\frac{2\pi}{n}). Since (n = 3), and (T=\frac{2\pi}{B}), then (\frac{2\pi}{B}=\frac{2\pi}{3}), so (B = 3).

Step3: Determine the sign

The sine function (y=\sin\theta) has a (y) - value of (0) at (\theta = 0) and is increasing near (\theta=0). The given function is decreasing near (\theta = 0). For (y = A\sin(B\theta)), when (A<0), the graph is reflected over the (x) - axis. Since (A=- 4) and (B = 3), the function is (y=-4\sin(3\theta))

Answer:

(y=-4\sin(3\theta)) (the third option (y = - 4\sin3\theta))