Q. 28 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 21–30, match the given function to one of the graphs (A)– (J).  

Step-by-Step Solution

Answer

The graph of the given function is graph (G)  

Step 1. Given information

The trigonometric function is: $y=-2\cos \left(\frac{\mathrm{\pi }}{2}x\right)$

Step 2. To determine the amplitude and period of the sinusoidal function.

By comparing given function $-2\cos \left(\frac{\mathrm{\pi }}{2}x\right)$ with standard cosine function: $A\cos \left(\omega x\right)$

We get amplitude $A=-2$

Time period is $\frac{2\mathrm{\pi }}{{\displaystyle \frac{\mathrm{\pi }}{2}}}=4$

So the graph of the given function will have time period 4 and amplitude -2

Step 3. To find value of function at y-axis.

At y-axis 

$x=0$

then  $$\begin{gathered} y=-2\cos \left(\frac{\mathrm{\pi }}{2}.0\right) \\ y=-2\cos 0 \\ y=-2.1 \\ y=-2 \end{gathered}$$

graph will pass through (0,-2)

Now look at the given graphs from A-J we can see that 

graph (G) has amplitude 2 , time period  4 and passing point (0,-2)

So the graph for given function is graph (G)