Q. 65 - Precalculus Enhanced with Graphing Utilities

Question

Find an equation for each graph.

Step-by-Step Solution

Answer

Equation for the graph is:

$y=-3\cos \left(\frac{1}{2}x\right)$

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, $$\begin{gathered} y_{min}=-3 \\ {y}_{max}=3 \end{gathered}$$

$$\begin{gathered} A=\frac{1}{2}(y_{max}-y_{min}) \\ A=\frac{1}{2}(3-(-3\left)\right) \\ A=\frac{1}{2}\left(6\right) \\ A=3 \end{gathered}$$

Using the maximum at  $x=0$, the next maximum occurs at $x=4\mathrm{\pi }$

Therefore,

$T=4\mathrm{\pi }$

Step 3. Solve for the equation

$$\begin{gathered} \omega =\frac{2\mathrm{\pi }}{T} \\ \omega =\frac{2\mathrm{\pi }}{4\mathrm{\pi }} \\ \omega =\frac{1}{2} \end{gathered}$$

Therefore, the equation of the graph is 

$y=-3\cos \left(\frac{1}{2}x\right)$