Modeling w/Trig Functions Homework

This graph models the height of Jesse's yo-yo over time by using a trigonometric function. He starts timing the yo-yo when it is at its lowest position of 5 inches above the ground. The string length is 30 inches and it took 2.5 seconds to return to the same position.

Explain why the graph has a peak at 35 inches.

This graph models the height of Jesse's yo-yo over time by using a trigonometric function. He starts timing the yo-yo when it is at its lowest position of 5 inches above the ground. The string length is 30 inches and it took 2.5 seconds to return to the same position.

What is the amplitude of the function?

This graph models the height of Jesse's yo-yo over time by using a trigonometric function. He starts timing the yo-yo when it is at its lowest position of 5 inches above the ground. The string length is 30 inches and it took 2.5 seconds to return to the same position.

In order to find the vertical shift, we need to find the middle point between the high and low points. What is the vertical shift?

This graph models the height of Jesse's yo-yo over time by using a trigonometric function. He starts timing the yo-yo when it is at its lowest position of 5 inches above the ground. The string length is 30 inches and it took 2.5 seconds to return to the same position.

Recall the relationships $$p=\frac{2\pi}{b}$$  and $$b=\frac{2\pi}{p}$$  . Find the angular frequency (the b value) for the graph.

This graph models the height of Jesse's yo-yo over time by using a trigonometric function. He starts timing the yo-yo when it is at its lowest position of 5 inches above the ground. The string length is 30 inches and it took 2.5 seconds to return to the same position.

This graph can be modeled WITHOUT using a horizontal shift. Which function can be used as a basis for this graph?

This graph models the height of Jesse's yo-yo over time by using a trigonometric function. He starts timing the yo-yo when it is at its lowest position of 5 inches above the ground. The string length is 30 inches and it took 2.5 seconds to return to the same position.

Put the above observations together to find a function that will model the height of the yo-yo over time.

$$y=-A\cos\left(Bx\right)+D$$  

Fill in the blanks and type your resulting equation if

A=15, B=4pi/5, no shift, and D=20

(a)  

An ecologist studying a species of water beetle estimates the population of a colony over an eight week period. If t is the number of weeks after the initial estimate is made, then the population in thousands can be modeled by

$$P\left(t\right)=5+2\sin\left(\frac{\pi t}{3}\right)$$  , where $$0\le t\le8$$  .

What is the initial population?

An ecologist studying a species of water beetle estimates the population of a colony over an eight week period. If t is the number of weeks after the initial estimate is made, then the population in thousands can be modeled by

$$P\left(t\right)=5+2\sin\left(\frac{\pi t}{3}\right)$$  , where $$0\le t\le8$$  .

What are the smallest and largest populations?

An ecologist studying a species of water beetle estimates the population of a colony over an eight week period. If t is the number of weeks after the initial estimate is made, then the population in thousands can be modeled by

$$P\left(t\right)=5+2\sin\left(\frac{\pi t}{3}\right)$$  , where $$0\le t\le8$$  .

During what time interval(s) did the population exceed 6,000?