Pre-Calculus Test Questions: Applications of Sinusoidal Functions.

📚 Quick Study Guide

• 📈 Amplitude: The amplitude ($A$) is half the distance between the maximum and minimum values of the function. Represented as $y = A\sin(Bx - C) + D$ or $y = A\cos(Bx - C) + D$.
• 🔄 Period: The period ($P$) is the length of one complete cycle of the function. Calculated as $P = \frac{2\pi}{|B|}$.
• phase shift: The horizontal shift of the function. Given by $ \frac{C}{B}$
• vertical shift: The vertical shift is given by $D$
• 🌊 Sinusoidal Functions: Functions of the form $y = A\sin(Bx - C) + D$ and $y = A\cos(Bx - C) + D$, where:
  • $A$ = Amplitude
  • $B$ affects the period
  • $C$ is phase shift
  • $D$ = Vertical Shift
• 📐 Frequency: The number of cycles completed per unit of time. Frequency = $\frac{1}{Period} = \frac{|B|}{2\pi}$.
• ✍🏽 Modeling with Sinusoids:
  • Identify the maximum and minimum values.
  • Calculate the amplitude: $A = \frac{Maximum - Minimum}{2}$.
  • Determine the vertical shift: $D = \frac{Maximum + Minimum}{2}$.
  • Find the period from the given information.
  • Calculate $B$ using $B = \frac{2\pi}{Period}$.
  • Write the equation in the form $y = A\sin(Bx - C) + D$ or $y = A\cos(Bx - C) + D$.

Practice Quiz

1. A Ferris wheel with a radius of 25 meters completes one rotation every 40 seconds. What is the period of the sinusoidal function that models a rider's height above the ground?
   1. 10 seconds
   2. 20 seconds
   3. 30 seconds
   4. 40 seconds
2. The average daily temperature in a city can be modeled by a sinusoidal function. If the highest average temperature is 80°F and the lowest is 40°F, what is the amplitude of the function?
   1. 20°F
   2. 40°F
   3. 60°F
   4. 80°F
3. A buoy oscillates up and down as waves pass. If the distance between the highest and lowest point is 6 feet, and it takes 8 seconds for the buoy to go from its highest to its lowest point, what is the period of the buoy's motion?
   1. 4 seconds
   2. 8 seconds
   3. 12 seconds
   4. 16 seconds
4. A pendulum swings back and forth, and its horizontal displacement can be modeled by a sinusoidal function. If the maximum displacement from the center is 10 cm, and it takes 2 seconds to complete one full swing, what is the equation of the sinusoidal function, assuming it starts at the center?
   1. $y = 10\sin(\pi t)$
   2. $y = 10\cos(\pi t)$
   3. $y = 20\sin(\pi t)$
   4. $y = 20\cos(\pi t)$
5. The water depth in a harbor varies sinusoidally with time. On a certain day, the high tide is at 8 feet and the low tide is at 2 feet. If the time between high and low tide is 6 hours, what is the vertical shift of the sinusoidal function?
   1. 2 feet
   2. 3 feet
   3. 5 feet
   4. 8 feet
6. A sound wave can be modeled by a sinusoidal function. If the frequency of the sound wave is 440 Hz, what is the period of the wave?
   1. $\frac{1}{440}$ seconds
   2. 440 seconds
   3. $2\pi \cdot 440$ seconds
   4. $\frac{440}{2\pi}$ seconds
7. The voltage in an AC circuit varies sinusoidally with time. If the peak voltage is 170 volts and the frequency is 60 Hz, what is the equation of the sinusoidal function, assuming the voltage starts at 0?
   1. $V(t) = 170\sin(60t)$
   2. $V(t) = 170\cos(60t)$
   3. $V(t) = 170\sin(120\pi t)$
   4. $V(t) = 170\cos(120\pi t)$