Q. 70 - Precalculus Enhanced with Graphing Utilities

Question

Find an equation for each graph.

Step-by-Step Solution

Answer

Equation for the graph is:

$y=-\cos \left(\mathrm{\pi x}\right)$

Step 1. Inspecting the y-intercept

Referring to the given graph the type of sinusoid can be determined immediately by inspecting its y- intercept if its at a maximum(highest point) and a minimum(lowest point) or neither

Since the maximum is at its y-intercept, then the graph is a cosine function and hence, $A<0$.

Step 2. Find the amplitude and period

From the graph, $$\begin{gathered} y_{min}=-\mathrm{\pi } \\ {y}_{max}=\mathrm{\pi } \end{gathered}$$

$$\begin{gathered} A=-\frac{1}{2}(y_{max}-y_{min}) \\ A=-\frac{1}{2}(\mathrm{\pi }-(-\mathrm{\pi }\left)\right) \\ A=-\frac{1}{2}\left(2\mathrm{\pi }\right) \\ A=-\mathrm{\pi } \\ \left|\mathrm{A}\right|=\mathrm{\pi } \end{gathered}$$

Using the maximum at  $x=0$, the next maximum occurs at $x=2\mathrm{\pi }$

Therefore,

$T=2\mathrm{\pi }$

Step 3. Solve for the equation

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

Therefore, the equation of the graph is 

$y=-\cos \left(\mathrm{\pi x}\right)$