1. Warm Up
Use the function y = 4 sin(7x) + 9 to answer the following questions:
What is the vertical distance between the lowest and highest points on the curve?
When x = 0, what is the value of y?
When x = π/14, what is the value of y?
When x = π/6, what is the value of y?

2. Trigonometric
Word Problems
Unit 6

3. Modeling with Trig
Many real world scenarios can be modeled with the sine and cosine curves. These movements are said to be sinusoidal
The height of a Ferris Wheel
Populations of Predators and Prey
A buoy bobbing up and down
Temperatures

4. Predator and Prey
In the wild, predators such as wolves need prey such as sheep to survive. The population of the wolves and the sheep are cyclic in nature.
If there are more sheep, more wolves can survive. But as more wolves exist, more sheep are eaten, thus the sheep population goes down. This causes the wolf population to also go down. But as few wolves exist, the sheep can start growing again. And thus…a cycle.

5. Wolves and Sheep Suppose the population of the wolves W is modeled by
W(m)= sin(π/6∙m)
and population of the sheep S is modeled by
S(m)=10, cos(π/6∙m)
where m is the time in months.

7. W(m)=3000+1000sin(π/6∙m) S(m)=10,000+5000cos(π/6∙m)
Describe the transformation of the sine function to the Wolf Population W(m).
Amplitude:
Vertical Translation:
Period:
Describe the transformation of the cosine function to the Sheep Population S(m).

8. W(m)=3000+1000sin(π/6∙m) S(m)=10,000+5000cos(π/6∙m)
What are the maximum number and minimum number of wolves?
What are the maximum number and minimum number of sheep?
During which months does the wolf population reach a maximum? The sheep?

9. Bobbing Buoy
A buoy in the harbor of San Juan, Puerto Rico, bobs up and down. The distance between the highest and lowest point is 4 feet. It moves from its highest point down to its lowest point and back to its highest point every 8 seconds. 
Find the equation of the motion for the buoy assuming that it is at its equilibrium point at t=0 and the buoy is on its way down at that time.
Determine the height of the buoy at 2 seconds.
Determine the height of the buoy at 12 seconds.
Y = -2sin(pi/4*t)
2ft
0ft (at equilibrium)

10. Riding a Ferris Wheel
At the carnival you decide to ride the Ferris Wheel. The wheel is 3ft off of the ground and has a diameter of 38ft. Once loaded, the wheel makes a revolution every 12 seconds.
Draw a graph and write a function to model the Ferris Wheel
How high would you be in
4 seconds?
Y = -19cos(pi/6*t)+22
Y = -19cos(pi/6*4)+22 = -19cos(2pi/3)+22 = -19(-1/2)+22 = 31.5 ft
22 = -19cos(pi/6*t)+22  0 = -19cos(pi/6*t)  0 = cos(pi/6*t)  When is cos = 0? (aka, when is the x-coord = 0 on the unit circle)  Need cos(pi/2) and cos(3pi/2)  t = 3, 9, 15… (t = 3 + 6n)

11. Homework
Complete the worksheet