Pre-Calculus examples: Using trig graphs to model tides and temperature.

📚 Quick Study Guide

• 🌊 Tides: Tides follow a roughly sinusoidal pattern, meaning we can use sine or cosine functions to model their height over time. The general form is $h(t) = A\cos(B(t - C)) + D$, where $h(t)$ is the height of the tide at time $t$, $A$ is the amplitude, $B$ is related to the period, $C$ is the horizontal shift, and $D$ is the vertical shift.
• 🌡️ Temperature: Similar to tides, daily or seasonal temperature variations can be modeled using sinusoidal functions. The amplitude represents the difference between the average temperature and the highest or lowest temperature. The period is the length of one full cycle (e.g., 365 days for yearly temperature).
• 📈 Amplitude (A): Represents half the difference between the maximum and minimum values of the function. $A = \frac{max - min}{2}$.
• ⏳ Period: The length of one complete cycle. If the function is $f(x) = \cos(Bx)$ or $f(x) = \sin(Bx)$, the period is given by $\frac{2\pi}{B}$.
• ↔️ Horizontal Shift (C): Shifts the graph left or right. It determines the starting point of the cycle.
• ↕️ Vertical Shift (D): Shifts the graph up or down. Represents the midline (average value) of the function. $D = \frac{max + min}{2}$.

Practice Quiz

1. What parameter in the sinusoidal function affects the height of the wave (tide or temperature)?
   1. Amplitude
   2. Period
   3. Horizontal Shift
   4. Vertical Shift
2. If the high tide is 8 feet and the low tide is 2 feet, what is the amplitude of the sinusoidal function modeling the tides?
   1. 2 feet
   2. 3 feet
   3. 5 feet
   4. 6 feet
3. What does the vertical shift (D) represent in the context of modeling temperature variations?
   1. The difference between the highest and lowest temperatures
   2. The average temperature
   3. The period of temperature change
   4. The rate of temperature change
4. The temperature in a city varies sinusoidally over the year. The maximum temperature is 90°F and the minimum is 50°F. What is the midline (vertical shift)?
   1. 20°F
   2. 40°F
   3. 70°F
   4. 90°F
5. A tide cycle completes approximately every 12 hours. What is the value of $B$ in the equation $h(t) = A\cos(B(t - C)) + D$ if $t$ is measured in hours?
   1. $\pi/12$
   2. $\pi/6$
   3. $\pi$
   4. $2\pi$
6. Which transformation of the cosine function, $f(x) = A\cos(B(x-C))+D$, affects the starting point of the cycle?
   1. A (Amplitude)
   2. B (Period)
   3. C (Horizontal Shift)
   4. D (Vertical Shift)
7. If a sinusoidal function models the daily temperature, and you want to shift the graph to the right by 3 hours, what parameter would you adjust?
   1. Amplitude
   2. Period
   3. Horizontal Shift
   4. Vertical Shift