Trigonometry / Unit 4: Graphs of Trigonometric Functions
4.2 Amplitude and Period
Notes
For $y = A\sin(Bx)$ or $y = A\cos(Bx)$:
- Amplitude $= |A|$: the distance from the midline to a peak. A negative $A$ flips the graph upside down.
- Period $\displaystyle = \frac{2\pi}{|B|}$: the length of one full cycle. A larger $B$ squeezes the wave.
- Example: $y = -3\sin(2x)$ has amplitude 3, period $\displaystyle \frac{2\pi}{2} = \pi$, and is reflected over the $x$-axis.
- The five key points of one cycle are equally spaced, one quarter of a period apart.
Practice
What is the amplitude of $y = -3\sin(2x)$?
Amplitude is $|A| = |-3| = 3$.
What is the period of $y = \cos(4x)$?
Period $\displaystyle = \frac{2\pi}{4} = \frac{\pi}{2}$.
What is the period of $\displaystyle y = \sin\left(\frac{x}{3}\right)$?
Here $\displaystyle B = \frac{1}{3}$, so period $\displaystyle = 2\pi \div \frac{1}{3} = 6\pi$.
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