Question

Draw a graph of cosine function with an amplitude of $\frac{7}{8}$ and a period of $\frac{2\pi }{5}$. Then write an equation for the cosine function.

Short answer

The equation of the cosine function is $y=\frac{7}{8}\cos \left(5x\right)$.

The amplitude of$y=\frac{7}{8}\cos \left(5x\right)$ is$\frac{7}{8}$.

The period of $y=\frac{7}{8}\cos \left(5x\right)$ is$\frac{2\pi }{5}$.

Step-by-step solution

Step 1. Write down the given information.

The given cosine function has an amplitude of $\frac{7}{8}$ and a period of $\frac{2\pi }{5}$.

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 a period of $\frac{360°}{\left|b\right|}\text{or}\frac{2\pi }{\left|b\right|}$.

Step 3. Write the equation of sine function of given amplitude and period.

Since the given cosine function has an amplitude of $\frac{7}{8}$ and a period of $\frac{2\pi }{5}$. Therefore, plugging $\left|a\right|=\frac{7}{8}$in cosine function $y=a\cos \left(bx\right)$ and equating period of cosine function $\frac{2\pi }{\left|b\right|}=\frac{2\pi }{5}$ to get $b=5$.

Now, plugging $\left|a\right|=\frac{7}{8}$ and $b=5$ in $y=a\cos \left(bx\right)$. Therefore, the equation of cosine function of given amplitude and period is written as $y=\frac{7}{8}\cos \left(5x\right)$.

Step 4. Sketch the graph for the obtained function.

The graph for the function $y=\frac{7}{8}\cos \left(5x\right)$ of amplitude $\left|a\right|=\frac{7}{8}$ and period $\frac{2\pi }{5}$ is shown below.

Step 5. Interpretation from the graph.

The equation of the cosine function is $y=\frac{7}{8}\cos \left(5x\right)$.

The amplitude of $y=\frac{7}{8}\cos \left(5x\right)$is $\frac{7}{8}$.

The period of $y=\frac{7}{8}\cos \left(5x\right)$ is $\frac{2\pi }{5}$.