Chapter 13.5, Problem 19PPSE

Solution

To write: A equation of cosine function whose graph is shown below.

  

The equation of cosine curve is written as $y=-3\cos \left(2\theta \right)$ where $a<0$ .

Given:

The graph of the cosine function is shown below.

  

Concept used:

The cosine curve is of the form $y=a\cos \left(b\theta \right)$ where $a\ne 0$ and $b>0$ has an amplitude of $\left|a\right|$ , period $\left(T\right)=\frac{2\pi }{b}$ and number of cycles $\left(b\right)$ in the interval from $0\text{ to }2\pi$ . And use the five-point pattern of $\left(\theta \right)$ values which follow a pattern for $\left(y\right)$ -values as $\min ,0,\max ,0\text{ and }\min$ where the curve cuts the $\left(\theta \right)$ -values and the value of $a<0$ in the curve $y=a\cos \left(b\theta \right)$ .

Calculation:

In order to write the equation of cosine function, evaluate the missing parameters using the concept stated above. Therefore, the maximum and minimum values of the function are $3\text{ and }-3$ respectively.

And the curve repeats itself after the interval from $0\text{ to }\pi$ . Therefore, the period of the function is $\pi$ .

And using the result for period of the function.

  $\begin{array}{c}\text{Period}\left(T\right)=\pi \text{ gives,}\\ \frac{2\pi }{b}=\pi \\ b=2\end{array}$

Now $\left(b\right)=2$ implies that the number of cycles of the function in the interval $0\text{ to }2\pi$ is $2$ .

And the curve of the graph starts from $-1$ which means $a=-3$ as $a<0$ . Hence, the equation of the curve using $a=-3$ and $b=2$ is written as $y=-3\cos \left(2\theta \right)$ .

Conclusion:

The equation of cosine curve is written as $y=-3\cos \left(2\theta \right)$ where $a<0$ .