Q. 100 - Precalculus Enhanced with Graphing Utilities

Question

Explain how you would scale the x-axis and y-axis before graphing $y=3\cos \left(\mathrm{\pi x}\right)$. 

Step-by-Step Solution

Answer

The x-axis will be scaled with multiple of $\frac{1}{2}$ units from the origin.

The y-axis will be scaled with multiple of $3$ units from origin.

Step 1. Concept used

If $\omega >0$, then the amplitude and period of $y=A\cos \omega x$ are given by

Amplitude$=\left|A\right|$

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

Step 2. Find the amplitude and period

Comparing the equation $y=3\cos \left(\mathrm{\pi x}\right)$ to $y=A\cos \left(\omega x\right)$ we find

$$\begin{gathered} A=3 \\ \omega =\mathrm{\pi } \end{gathered}$$

So the amplitude is given as

$\begin{array}{rcl}\left|A\right|& =& \left|3\right|\\ & =& 3\end{array}$

and the period is given as

$\begin{array}{rcl}\frac{2\mathrm{\pi }}{\omega }& =& \frac{2\mathrm{\pi }}{\mathrm{\pi }}\\ & =& 2\end{array}$

Step 3. Find the scale

As the amplitude of the function is $3$ so the $y$ axis will be scaled with the multiple of $3$ units from the origin.

The period of the function is $2$ so the $x$ axis will be scaled with the multiple of $\frac{1}{2}$ units from the origin.