Trigonometry / Unit 4: Graphs of Trigonometric Functions
4.4 Writing and Interpreting Sinusoidal Models
Notes
Many real situations (tides, Ferris wheels, daylight hours) repeat in a wave pattern.
- Midline $\displaystyle D = \frac{\max + \min}{2}$. Amplitude $\displaystyle |A| = \frac{\max - \min}{2}$.
- $\displaystyle B = \frac{2\pi}{\text{period}}$.
- Use cosine if the graph starts at a maximum, sine if it starts at the midline going up.
- Example: max 7, min $-1$ gives midline $3$ and amplitude 4.
Practice
A sinusoid has a maximum of 7 and a minimum of $-1$. What are its amplitude and midline?
Amplitude $\displaystyle = \frac{7 - (-1)}{2} = 4$ and midline $\displaystyle = \frac{7 + (-1)}{2} = 3$.
A cosine-shaped wave has a maximum of 5 at $x = 0$, a minimum of $-5$, and a period of $\pi$. Which equation fits?
Amplitude 5, and $\displaystyle B = \frac{2\pi}{\pi} = 2$. It starts at a maximum, so use cosine.
Tide height in feet is $\displaystyle h(t) = 4\cos\left(\frac{\pi t}{6}\right) + 6$, where $t$ is in hours. What is the minimum height?
The minimum is the midline minus the amplitude: $6 - 4 = 2$.
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