Q. 11 - 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. 

$y=3\cos \left(\mathrm{\pi x}-2\right)+5$

Step-by-Step Solution

Answer

Amplitude, $A=3$

Time 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 cosine function form is given as: 

$y=3\cos \left(\mathrm{\pi x}-2\right)+5$

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:

Amplitude:

$$\begin{gathered} A=\left|3\right| \\ =3 \end{gathered}$$

Time period: 

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

Phase shift:

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

Step 2. Graph the function

The graph is given as: