Q. 103 - Precalculus Enhanced with Graphing Utilities

Question

Explain how the amplitude and period of a sinusoidal graph are used to establish the scale on each coordinate axis. 

Step-by-Step Solution

Answer

The amplitude is of a sinusoidal graph is used to scale the y-axis and the period scales the x-axis. 

For a sinusoidal function of the form $y=A\sin \left(\omega x\right)$ the $y$ axis is scaled by a factor of $A$ and the $x$ axis is scaled by a factor of $\frac{1}{\omega }$.

Step 1. Concept Used

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

Amplitude$=\left|A\right|$

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

Step 2. Find what amplitude and period represents

The amplitude of a sinusoidal function represents the range of the sinusoidal function. 

So for the sinusoidal curve $y=A\sin \left(\omega x\right)$ and $y=A\cos \left(\omega x\right)$ the amplitude $\left|A\right|$ means that the graph of the function lies between the values $-A$ and $A$.

The period of the function is given as $T=\frac{2\mathrm{\pi }}{\omega }$ represents the length of the one cycle and after $\frac{2\mathrm{\pi }}{\omega }$ the graph starts repeating.

One cycle is represented by the interval $\left[0,\frac{2\mathrm{\pi }}{\omega }\right]$.

Step 3. Find the scale in each axis

As the y value lies between $-A$ and $A$, so the y-axis is scaled by a factor of $A$.

As one interval of the cycle is given as $\left[0,\frac{2\mathrm{\pi }}{\omega }\right]$ so the x-axis is scaled by a factor of $\frac{1}{\omega }$.