Q. 74 - Precalculus Enhanced with Graphing Utilities

Question

Find an equation for the graph.

Step-by-Step Solution

Answer

The equation of the given graph is $y=-2\cos \left(\mathrm{\pi x}\right)$.

Step 1. Determine the sinusoidal function.

Consider the given graph,

The type of sinusoidal graph can be determined immediately by inspecting its y-intercept if its at a maximum(highest point) and a minimum(lowest point) or neither.

As the minimum is at its y-intercept, then the graph is a cosine function and$A<0$. Then,

$$\begin{gathered} y_{max}=2 \\ {y}_{min}=-2 \end{gathered}$$

Step 2. Find the amplitude and period.

Consider the given question,

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

The period is $2$. So,

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

Step 3. Form the equation.

Vertical shift,

$$\begin{gathered} b=\frac{1}{2}\left(y_{max}+y_{min}\right) \\ b=\frac{1}{2}\left(2-2\right) \\ b=0 \end{gathered}$$

Consider the form of the function,

$y=A\sin \left(\omega x\right)+b$

Substitute the values,

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