Q. 38 - Precalculus Enhanced with Graphing Utilities

Question

n 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  

$y=-3\cos x$

Step-by-Step Solution

Answer

The graph is

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

Range is $\left[-3,3\right]$

Step 1. Given information

The function to be plotted is 

$y=-\cos x$

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

By comparing the given function $y=-3\cos x$

with $y=A\cos \left(\omega x\right)$

we get amplitude:$\left|A\right|=3$

 time period 

$$\begin{gathered} T=\frac{2\mathrm{\pi }}{\omega } \\ T=\frac{2\mathrm{\pi }}{1} \\ T=2\mathrm{\pi } \end{gathered}$$

The graph will lie  between -3 and 3 on the y-axis. One cycle begins at $0$ and ends at $2\mathrm{\pi }$

Step 3. : Divide the interval 2 π into four subintervals of the same length.

Divide the interval $2\mathrm{\pi }$ into four subintervals,

each of length 

$\frac{2\mathrm{\pi }}{4}=\frac{\mathrm{\pi }}{2}$

The x-coordinates of the five key points are :

First x-coordinate is 0

second x-coordinate is $0+\frac{\mathrm{\pi }}{2}=\frac{\mathrm{\pi }}{2}$

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

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

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

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 $y=-3\cos x$

 multiply the y-coordinates of the five key points for $\cos x$ by -3.

 The five key points on the graph are 

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

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  $\frac{\mathrm{\pi }}{2}$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 -3 to 3

So range of the function is $\left[-3,3\right]$