1. Starting off the day with a problem… The Bolivar Lighthouse is located on a small island 350ft from the shore of the mainland as shown in the figure. x d 350 ft (a) Express the distance d as a function of the angle x. (b) If x is 1.55 rad, what is d ? ft

2. And another one just for good measure A hot-air balloon is being blown due east from point P and traveling at a constant height of 800ft. The angle y is formed by the ground and the line of vision P to the balloon. This angle changes as the balloon travels. x y 800 ft P (a) Express the horizontal distance x as a function of the angle y. (b) When the angle is rad, what is its horizontal distance from P ? ft (c) An angle of rad is equivalent to how many degrees?

3. The beginning of Section 4.6a

4. Example 1: Combining the sine function with Graph each of the following functions for, adjusting the vertical window as needed. Which of the functions appear to be periodic? First, put all four functions into your calculator, then you can turn their graphs on and off as needed… Vertical window [–10, 20] [–25, 25] [–1.5, 1.5] Only the graph of this third function exhibits periodic behavior over the given interval!!!

5. Example 2: Verifying periodicity algebraically Verify algebraically that is periodic and determine its period graphically. It follows that First, recall the period of the basic sine function Next, recall the fact thatfor all x. (Can you explain why this is true???) By the periodicity of sine Check the graph  What does the period appear to be? Period =

6. More Guided Practice Graph the given functions for, adjusting the vertical window as needed. State whether or not the functions appears to be periodic. 1. Graph window: Not Periodic 2. Graph window: Not Periodic 3. Graph window: Periodic

7. More Guided Practice Verify algebraically that the given function is periodic and determine its period graphically. Sketch a graph showing two periods. 1. Since the period of cos(x) is, we have Graph window:

8. More Guided Practice Verify algebraically that the given function is periodic and determine its period graphically. Sketch a graph showing two periods. 2. Since the period of cos(x) is, we have Graph window:

9. Practice Problems Prove algebraically that is periodic and find the period graphically. State the domain and range and sketch a graph showing two periods. First, a reminder note regarding notation: A function like is more frequently written as (but this shorthand notation will not be recognized by a calculator)

10. Practice Problems Prove algebraically that is periodic and find the period graphically. State the domain and range and sketch a graph showing two periods. To prove that the function is periodic, we need to show that for all x. Changing notation By periodicity of sine Changing notation

11. Practice Problems Prove algebraically that is periodic and find the period graphically. State the domain and range and sketch a graph showing two periods. Graph the function in the window: What does the period appear to be?  Period = Domain: Range: How does the graph of this function compare to that of the basic sine function??? (let’s graph both in the same window)

12. Practice Problems Find the domain, range, and period of each of the following functions. Sketch a graph showing four periods. Graph window: Wherever the basic tangent function is defined, f is also defined. Domain: All reals except odd multiples of The range of the basic tangent function is all reals, but f is always greater than or equal to zero. (why???) Range: The period of f is the same as the basic tangent function:

13. Practice Problems Find the domain, range, and period of each of the following functions. Sketch a graph showing four periods. Graph window: Wherever the basic sine function is defined, g is also defined. Range: The period of g is half that of the basic sine function: Domain: The range of the basic sine function is –1 to 1 (inclusive), but g is always greater than or equal to zero.

14. Practice Problems The graph of oscillates between two parallel lines. What are the equations of the two lines? The basic sine function oscillates between –1 and 1, so f(x) must oscillate between 0.5x – 1 and 0.5x + 1… The lines: To verify our answer visually, graph all three functions in Bonus question: Is f(x) periodic? Why or why not?  This function is not periodic!!!

15. Practice Problems State the domain and range of the given functions, and sketch a graph showing four periods. 1. Graph window: Domain:Range: 2. Graph window: Domain:Range: 3. Graph window: Domain: Range:

16. Practice Problems The graphs of the given functions oscillate between two parallel lines. Find the equations of the two lines, and graph each function in the same window with its respective lines. 1. Graph window: The lines: 2. Graph window: The lines: