Q. 9 - Precalculus Enhanced with Graphing Utilities

Question

Find the amplitude, period, and phase shift of each function. Graph each function. Be sure to label key points. Show at least two periods.   

$y=4\sin \left(\mathrm{\pi x}+2\right)-5$

Step-by-Step Solution

Answer

Amplitude, $A=4$

Tine period,$T=2$

Phase shift$=\frac{-2}{\mathrm{\pi }}$

The graph of the function is given as:

Step 1. Find the amplitude, period, and phase shift.

The general sine function form is

 given as: 

$y=A\sin \left(\omega t-\varphi \right)+B$

Where,

$A=$Amplitude

$\frac{2\mathrm{\pi }}{\omega }=$Time period

$\frac{\varphi }{\omega }=$Phase shift

Comparing the given function with the general function, we get:  

$$\begin{gathered} A=\left|4\right| \\ A=4 \end{gathered}$$

Time period,

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

Phase shift:

$\frac{\varphi }{\omega }=\frac{-2}{\mathrm{\pi }}$

Step 2. Graph the function

The graph is given as: