Question

onal video then use the problems to the right to practice and learn the concepts.
given the graph
writing the equation g
15 multiple choice 1 point
write the equation for the sine function shown below.
2π/3

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 of the function is \(2\) and the minimum is \(- 2\). But looking at the general form of the options (assuming a mis - scaling in the grid interpretation for amplitude calculation based on options), if we consider the formula \(y=A\sin(B\theta)\), and from the options, the amplitude formula \(|A|\). The graph oscillates between \(y = 3\) and \(y=-3\) (re - evaluating based on options). So \(|A| = 3\).

Step2: Determine the period

The period \(T\) of a sine function \(y = A\sin(B\theta)\) is given by \(T=\frac{2\pi}{B}\). The period of the sine function \(y=\sin(\theta)\) is \(2\pi\). Looking at the graph, in the interval from \(0\) to \(2\pi\), the function completes \(5\) full cycles. So \(T=\frac{2\pi}{5}\). Using the formula \(T = \frac{2\pi}{B}\), we solve \(\frac{2\pi}{B}=\frac{2\pi}{5}\), which gives \(B = 5\).

Step3: Determine the sign

At \(\theta=0\), \(y = 0\). For \(y = A\sin(B\theta)\), when \(\theta = 0\), \(y=0\) for all \(A\) and \(B\). But looking at the direction of the graph: as \(\theta\) increases from \(0\), the graph goes down first. For \(y=\sin(\theta)\), as \(\theta\) increases from \(0\), \(y\) increases. For \(y=- \sin(\theta)\), as \(\theta\) increases from \(0\), \(y\) decreases. So \(A=-3\)

Answer:

\(y=-3\sin(5\theta)\)