Answer:
Part A
The amplitude of a cosine function is the distance between its highest and lowest points. In this case, the highest point is 8 feet and the lowest point is 8 - 12 = -4 feet, so the amplitude is 8 - (-4) = 12 feet.
The period of a cosine function is the time it takes for the function to complete one cycle of its movement. In this case, the bird completes one cycle of its movement every 16 seconds, so the period is 16 seconds.
Part B
The cosine function that could represent the situation is:
y = 6 cos(2πt / 16) + 8
This function has an amplitude of 6 feet and a period of 16 seconds. It also passes through the point (0, 8) when t = 0, which is the condition given in the problem.
Part C
The bird will reach its lowest height when the cosine function is equal to -6. This happens when 2πt / 16 = π / 2, or t = 4 seconds.
Here is a graph of the function:
[Image of a cosine function with an amplitude of 6 feet and a period of 16 seconds. The graph reaches its lowest point at t = 4 seconds.]
I hope this helps! Let me know if you have any other questions.
Explanation: