Q. 68 - Precalculus Enhanced with Graphing Utilities

Question

Find an equation for each graph.

Step-by-Step Solution

Answer

Equation for the graph is:

$y=-\frac{5}{2}\cos \left(\mathrm{\pi x}\right)$

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 minimum 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. $$\begin{gathered} y_{min}=-\frac{5}{4} \\ {y}_{max}=\frac{5}{4} \end{gathered}$$

$$\begin{gathered} A=\frac{1}{2}(y_{max}-y_{min}) \\ A=\frac{1}{2}(\frac{5}{4}-(-\frac{5}{4}\left)\right) \\ A=\frac{1}{2}\left(\frac{10}{4}\right) \\ A=\frac{5}{4} \end{gathered}$$

Using the maximum at  $x=0$, the next maximum occurs at $x=2$

Therefore,

$T=2$

Step 3. Solve for the equation

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

Therefore, the equation of the graph is 

$y=-\frac{5}{2}\cos \left(\mathrm{\pi x}\right)$