Write the equation of a sine function that has the given characteristics.
-Amplitude: 4
Period: $3 \pi$
A) $y = 6 \sin \left( \frac { 2 } { 5 } x \right)$
B) $y = 5 \sin \left( \frac { 1 } { 3 } x \right)$
C) $y = 5 \sin ( 6 x )$
D) $y = \sin ( 6 x ) + 5$

Question

Quizplus · Accepted Answer

The general form of a sine function is $y = A \sin(B(x - C)) + D$, where $A$ is the amplitude, $B$ affects the period according to the formula $Period = \frac{2\pi}{|B|}$, $C$ is the phase shift, and $D$ is the vertical shift. For an amplitude of 4 and a period of $3\pi$, we look for a function with $A = 4$ and $B$ such that $Period = \frac{2\pi}{|B|} = 3\pi$, which simplifies to $|B| = \frac{2}{3}$. Among the options, only choice B has the correct form, $y = 5 \sin \left( \frac { 1 } { 3 } x \right)$, which matches the amplitude requirement incorrectly, indicating a mistake in the explanation regarding amplitude. However, it correctly matches the period requirement based on the given options, suggesting a possible error in the question's details or in the interpretation of the amplitude.

Write the Equation of a Sine Function That Has the Given