Q. 12 - 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. 

$y=2\cos \left(2\mathrm{\pi x}-4\right)-1$

Step-by-Step Solution

Answer

Amplitude, $A=2$

Time period, $T=1$

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 :

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

Where, 

$A=$Amplitude

$T=$Time period

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

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

Amplitude:

$$\begin{gathered} A=\left|2\right| \\ A=2 \end{gathered}$$

Time period:

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

Phase shift:

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

Step 2. Graph the function

The graph of the function is given as: