Question

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

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 \(y = - 4\sin3\theta\))