Q. 45 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 35–58, graph each function. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function.   

$y=-\frac{1}{2}\cos \left(2x\right)$

Step-by-Step Solution

Answer

The graph of the function is:

The domain is $\left(-\infty ,\infty \right)$.

The range is $\left[-\frac{1}{2},\frac{1}{2}\right]$.

Step 1. Given information

The function to be plotted is :

$y=-\frac{1}{2}\cos \left(2x\right)$

Step 2. Determine the amplitude and period of the sinusoidal function.

By comparing the given function $y=-\frac{1}{2}\cos \left(2x\right)$ with $A\cos \left(\omega x\right)$

we get amplitude: $\left|A\right|=\frac{1}{2}$

Time period : $$\begin{gathered} T=\frac{2\mathrm{\pi }}{\omega } \\ T=\frac{2\mathrm{\pi }}{2} \\ T=\mathrm{\pi } \end{gathered}$$

The graph will lie  between $-\frac{1}{2}$ and $\frac{1}{2}$ on the y-axis. One cycle begins at  $x=0$ and ends at $x=\mathrm{\pi }$.

Step 3. : Divide the interval into four subintervals of the same length.

Divide the interval $\left[0,\mathrm{\pi }\right]$ into four subintervals,

each of length:$\frac{\mathrm{\pi }}{4}$

 The x-coordinates of the five key points are :

First x-coordinate is 0

second x-coordinate is $0+\frac{\mathrm{\pi }}{4}=\frac{\mathrm{\pi }}{4}$

Third x-coordinate is $\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\pi }}{4}=\frac{\mathrm{\pi }}{2}$

Fourth x-coordinate is $\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{4}=\frac{3\mathrm{\pi }}{4}$

Fifth x-coordinate is $\frac{3\mathrm{\pi }}{4}+\frac{\mathrm{\pi }}{4}=\mathrm{\pi }$

These values represent the x-coordinates of the five key points on the graph. 

Step 4. Use the endpoints of these subintervals to obtain five key points on the graph.

Since $y=-\frac{1}{2}\cos \left(2x\right)$

Hence, multiply the y-coordinates of the five key points for $y=\cos x$ by $-\frac{1}{2}$

The five key points on the graph are :

$\left(0,-\frac{1}{2}\right),\left(\frac{\mathrm{\pi }}{4},0\right),\left(\frac{\mathrm{\pi }}{2},\frac{1}{2}\right),\left(\frac{3\mathrm{\pi }}{4},0\right),\left(\mathrm{\pi },-\frac{1}{2}\right)$

Step 5. Plot the five key points and draw a sinusoidal graph to obtain the graph of one cycle. Extend the graph in each direction to make it complete.

Plot the five key points obtained in Step 4 and fill in the graph . Extend the graph in each direction to obtain the complete graph . Notice that additional key points appear every  $\frac{\mathrm{\pi }}{4}$ radian.  

Step 6. To find domain and range of the function

As we can see that the value of $x$ is set of all real number

So domain is $\left(-\infty ,\infty \right)$

The y- value of the function in the graph lies from $-\frac{1}{2}$ to $\frac{1}{2}$.

So range of the function is $\left[-\frac{1}{2},\frac{1}{2}\right]$.