1. Precalculus 1/9/2015 DO NOW/Bellwork: 1) Take a unit circle quiz 2) You have 10 minutes to complete AGENDA Unit circle quiz Sin and Cosine Transformations investigation. Wrap up Essential Question: How can the study of trigonometry be used in the real world? HOMEWORK: NONE Objectives: You will be able to graph and write sinusoidal functions.

2. Graphing Sine and Cosine Pre Calculus

3. Cosine Function Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1001cos x 0x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = cos x

4. Sine Function Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 0010sin x 0x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = sin x

5. Properties of Sine and Cosine Functions 6. The cycle repeats itself indefinitely in both directions of the x-axis. Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 1. The domain is the set of real numbers. 5. Each function cycles through all the values of the range over an x-interval of. 2. The range is the set of y values such that.

6. Transformations – A look back Let’s go back to the quadratic equation in graphing form: y = a(x – h) 2 + k If a < 0: reflection across the x axis |a| > 1: stretch; and |a| < 1: shrink (h, k) was the vertex (locator point) h gave us the horizontal shift k gave us the vertical shift

7. Transforming the Cosine (or Sine (Sinusoid)

8. Key Points Amplitude – increase the output by a factor of the amplitude Remember the amplitude is always positive so you have to apply any reflections y = 3 cos x 30-303cos x 0x 1001cos x 0x

9. y x Example: y = 3 cos x Example: Sketch the graph of y = 3 cos x on the interval [– , 4  ]. Partition the reference interval [0, 2  ] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. maxx-intminx-intmax 30-303 y = 3 cos x 22 0x (0, 3) (, 0) (, 3) (, –3)

10. Amplitude The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically. If 0 < |a| < 1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis. y x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x y = 2 sin x y = sin x

11. Key Points Amplitude – increase the output by a factor of the amplitude Remember the amplitude is always positive so you have to apply any reflections y = 3 cos x 30-303cos x 0x 1001cos x 0x

12. y x Example: y = 3 cos x Example: Sketch the graph of y = 3 cos x on the interval [– , 4  ]. Partition the reference interval [0, 2  ] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. maxx-intminx-intmax 30-303 y = 3 cos x 22 0x (0, 3) (, 0) (, 3) (, –3)

13. Transforming the Cosine (or Sine) y = a (cos (bx – c)) + d Let’s look at d Just like in our other functions, d is the vertical shift, if d is positive, it goes up, if it is negative, it goes down. Graph y = cos x and y = cos x + 1 y = cos x and y = cos x – 2

14. Vertical Shift y = cos x + 1 Begin with y = cos x Add d to the output to adjust the graph Then shift up one unit y x 1001cos x 0x 21012 0x

15. Transforming the Cosine (or Sine) y = a (cos (bx – c)) + d Let’s look at c Just like in our other functions, c is the horizontal shift, if c is positive, it goes right, if it is negative, it goes left. If there is no b present, it is the same as other functions Graph: y = cos x and y = cos (x + π / 2 ) y = cos x and y = cos (x - π / 2 )

16. Horizontal shift y = cos (x + π / 2 ) Begin with y = cos x Now you must translate the input… the angle Then shift left π / 2 units y x 1001cos x 0x 1001cos x 0 x

17. Transforming Cosine (or Sine) y = a (cos (bx – c)) + d Let’s look at b Graph: y =cos x and y =cos 2x (b = 2) y =cos x and y =cos ½ x (b = ½ ) What happened?

18. Transforming the Period b has an effect on the period (normal is 2 π ) If b > 1, the period is shorter, in other words, a complete cycle occurs in a shorter interval If b < 1, the period is longer or a cycle completes over an interval greater than 2 π To determine the new period = 2π/b

19. y x Period of a Function period: 2 period: The period of a function is the x interval needed for the function to complete one cycle. For b  0, the period of y = a sin bx is. For b  0, the period of y = a cos bx is also. If 0 < b < 1, the graph of the function is stretched horizontally. If b > 1, the graph of the function is shrunk horizontally. y x period: 2 period: 4

20. Summarizing … Standard form of the equations: y = a cos (b(x – c)) + d y = a sin (b(x – c)) + d “a” - |a| is called the amplitude, like our other functions it is like a stretch it affects “y” or the output If a < 0 it also causes a reflection across the x- axis “d” – vertical shift, it affects “y” or the output “c” – horizontal shift (phase shift), it affects “x” or “ θ ” or the input “b” – period change “squishes” or “stretches out” the graph

21. Notice we didn’t write it as 6sin(10x-2) that would represent a phase shift of 2/10 not 2.

22. Slide 4.4 - 22 Example Combining a Phase Shift with a Period Change

23. Slide 4.4 - 23 Example Combining a Phase Shift with a Period Change

24. Slide 4.4 - 24 Constructing a Sinusoidal Model using Time

25. Slide 4.4 - 25 Constructing a Sinusoidal Model using Time

26. Real World Problem On a particular day in Marathon, FL high tide occurred at 9:36am. At that time the water at the end of the 33 rd Street boat ramp was 2.7 meters deep. Low tide occurred at 3:48 pm at which time the tide was only 2.1 meters deep. Assume the depth of the water is a sinusoidal function of time with a period of 12hours and 24 minutes. a) At what time on that day did the first low tide occur? b) What was the approximate depth of the water at 6am and 3pm that day? c) What was the 1 st time on that day when the water was 2.4 meters deep?