Q27. - Algebra 2

Question

Find the amplitude if it exists and period of the function $y=6\sin \left(\frac{2\theta }{3}\right)$ and then graph the function.

Step-by-Step Solution

Answer

The amplitude of $y=6\sin \left(\frac{2\theta }{3}\right)$ is 6.

The period of $y=6\sin \left(\frac{2\theta }{3}\right)$ is $3\pi$.

Step 1. Write down the given information.

The given function is $y=6\sin \left(\frac{2\theta }{3}\right)$.

Step 2. Concept used.

A function of the form $y=a\sin \left(bx\right) \text{and }y=a\cos \left(bx\right)$ has amplitude of $\left|a\right|$ and period $\frac{360°}{\left|b\right|} \text{or }\frac{2\pi }{\left|b\right|}$.

Step 3. Evaluating amplitude and period of the given function.

With the help of concept stated above, the amplitude and period of the function is evaluated as:

The amplitude of $y=6\sin \left(\frac{2\theta }{3}\right)$ is $\left|6\right|=6$.

The period of $y=6\sin \left(\frac{2\theta }{3}\right)$ is $\frac{2\pi }{\left|\frac{2}{3}\right|}=3\pi$.

Step 4. Sketch the graph for the function.

The graph for the function $y=6\sin \left(\frac{2\theta }{3}\right)$ is shown below.

Step 5. Conclusion.

The amplitude of $y=6\sin \left(\frac{2\theta }{3}\right)$ is 6.

The period of $y=6\sin \left(\frac{2\theta }{3}\right)$ is $3\pi$.