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Describe how the amplitude, period, and frequency of a sine function are related to its graph. Provide an example with a specific sine function.
Explain how to determine the amplitude and period of the cosine function .
A periodic function has a period of and an amplitude of . Write a possible equation for if it is a sine function.
How does the frequency of a periodic function change if the period is halved? Provide an example with a specific function.
Given the function , identify the amplitude, period, and phase shift.
Describe how the day-night cycle can be modeled using a trigonometric function. Include the key features of the function such as amplitude, period, and phase shift.
Explain how seasonal changes can be modeled using a trigonometric function. Include the key features of the function such as amplitude, period, and phase shift.
How can the concept of periodic behavior be applied to model the tidal patterns in oceans using trigonometric functions?
Discuss how the concept of periodic behavior can be used to model the phases of the moon using trigonometric functions.
Explain how the concept of periodic behavior can be used to model the annual migration patterns of birds using trigonometric functions.
A Ferris wheel has a diameter of 50 meters and completes one full rotation every 4 minutes. Write a trigonometric function that models the height of a passenger above the ground as a function of time in minutes, assuming the passenger starts at the lowest point.
A sound wave can be modeled by the function , where is the displacement and is the time in seconds. Determine the amplitude, period, and frequency of the sound wave.
The average daily temperature in a city can be modeled by the function , where is the temperature in degrees Celsius and is the time in months. Determine the highest and lowest average temperatures in the city.
A mass on a spring oscillates according to the function , where is the displacement in centimeters and is the time in seconds. Calculate the angular frequency, period, and amplitude of the oscillation.
The population of a certain species of fish in a lake oscillates seasonally and can be modeled by , where is the population and is the time in months. Determine the maximum and minimum population of the fish.
The average daily temperature in a city can be modeled by the function , where is the temperature in degrees Celsius and is the time in months. Determine the highest and lowest average temperatures in the city.
(Student response here)
The highest average temperature is degrees Celsius, and the lowest average temperature is degrees Celsius. These values are derived from the amplitude of the sine function, which is 10 degrees.