Q. 102 - Precalculus Enhanced with Graphing Utilities

Question

Explain the term period as it relates to the graph of a sinusoidal function. 

Step-by-Step Solution

Answer

The term period represents the value after which the graph of a sinusoidal function starts repeating itself.

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 period represents

The period of a cyclic function gives the x values after which the function starts repeating.

Sinusoidal functions of the form $y=A\sin \left(\omega x\right)$ or $y=A\cos \left(\omega x\right)$ where $\omega >0$ are cyclic functions and their period is denoted as $T=\frac{2\mathrm{\pi }}{\omega }$.

This means that the sinusoidal functions will start repeating after every $\frac{2\mathrm{\pi }}{\omega }$ , and its one cycle is given by the interval $\left[0,\frac{2\mathrm{\pi }}{\omega }\right]$.