Graphs of the Sine and Cosine Functions Lecture

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Lesson Objectives

Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions

1. www. Pinoy BIX .org Presents: GRAPHS OF SINE AND COSINE FUNCTIONS credit: Shawna Haider

2. GRAPHS OF SINE AND COSINE FUNCTIONS

3. We are interested in the graph of y = f ( x ) = sin x Start with a "t" chart and let's choose values from our unit circle and find the sine values. x y = sin x We are dealing with x 's and y 's on the unit circle to find values. These are completely different from the x 's and y 's used here for our function. x y 1 - 1 plot these points

4. y = f ( x ) = sin x choose more values x y = sin x If we continue picking values for x we will start to repeat since this is periodic. x y 1 - 1 plot these points join the points

5. Here is the graph y = f ( x ) = sin x showing from -2  to 6  . Notice it repeats with a period of 2  . It has a maximum of 1 and a minimum of -1 (remember that is the range of the sine function) 2  2  2  2 

6. What are the x intercepts? Where does sin x = 0? …-3  , -2  , -  , 0,  , 2  , 3  , 4  , . . . Where is the function maximum? Where does sin x = 1?

7. Where is the function minimum? Where does sin x = -1?

8. Thinking about transformations that you learned and knowing what y = sin x looks like, what do you suppose y = sin x + 2 looks like? The function value (or y value) is just moved up 2. y = sin x y = 2 + sin x This is often written with terms traded places so as not to confuse the 2 with part of sine function

9. Thinking about transformations that you've learned and knowing what y = sin x looks like, what do you suppose y = sin x - 1 looks like? The function value (or y value) is just moved down 1. y = sin x y = - 1 + sin x

10. Thinking about transformations that you learned and knowing what y = sin x looks like, what do you suppose y = sin ( x + /2) looks like? This is a horizontal shift by - /2 y = sin x y = sin ( x + /2)

11. Thinking about transformations that you learned and knowing what y = sin x looks like, what do you suppose y = - sin ( x )+1 looks like? This is a reflection about the x axis (shown in green) and then a vertical shift up one. y = sin x y = - sin x y = 1 - sin ( x )

12. What would the graph of y = f ( x ) = cos x l ook like? We could do a "t" chart and let's choose values from our unit circle and find the cosine values. x y = cos x We could have used the same values as we did for sine but picked ones that gave us easy values to plot. x y 1 - 1 plot these points

13. y = f ( x ) = cos x Choose more values. x y = cos x cosine will then repeat as you go another loop around the unit circle x y 1 - 1 plot these points

14. Here is the graph y = f ( x ) = cos x showing from -2  to 6. Notice it repeats with a period of 2. It has a maximum of 1 and a minimum of -1 (remember that is the range of the cosine function) 2  2  2  2 

15. Recall that an even function (which the cosine is) is symmetric with respect to the y axis as can be seen here

16. What are the x intercepts? Where does cos x = 0? …-4 , -2, , 0, 2, 4, . . . Where is the function maximum? Where does cos x = 1?

17. …-3 , -, , 3, . . . Where is the function minimum? Where does cos x = -1?

18. You could graph transformations of the cosine function the same way you've learned for other functions. Let's try y = 3 - cos ( x - /4) reflects over x axis moves up 3 moves right /4 y = cos x y = - cos x y = 3 - cos x y = 3 - cos ( x - /4)

19. What would happen if we multiply the function by a constant? y = 2 sin x All function values would be twice as high y = 2 sin x y = sin x The highest the graph goes (without a vertical shift) is called the amplitude . amplitude of this graph is 2 amplitude is here

20. For y = A cos x and y = A sin x ,  A  is the amplitude. y = 4 cos x y = -3 sin x What is the amplitude for the following? amplitude is 4 amplitude is 3

21. The last thing we want to see is what happens if we put a coefficient on the x . y = sin 2 x y = sin 2 x y = sin x It makes the graph "cycle" twice as fast. It does one complete cycle in half the time so the period becomes .

22. What do you think will happen to the graph if we put a fraction in front? y = sin 1/2 x y = sin x The period for one complete cycle is twice as long or 4 

23. So if we look at y = sin x the  affects the period. The period T = This will be true for cosine as well. What is the period of y = cos 4 x ? This means the graph will "cycle" every /2 or 4 times as often y = cos 4 x y = cos x

24. absolute value of this is the amplitude Period is 2  divided by this

25. Sample Problem Which of the following equations best describes the graph shown? (A) y = 3 sin (2 x ) - 1 (B) y = 2 sin ( 4x ) (C) y = 2 sin ( 2x ) - 1 (D) y = 4 sin ( 2x ) - 1 (E) y = 3 sin ( 4x ) -2p -1p 1p 2p 5 4 3 2 1 -1 -2 -3 -4 -5

26. Sample Problem Find the baseline between the high and low points. Graph is translated -1 vertically. Find height of each peak. Amplitude is 3 Count number of waves in 2  Frequency is 2 -2p -1p 1p 2p 5 4 3 2 1 -1 -2 -3 -4 -5 y = 3 sin( 2 x ) - 1

27. Which of the following equations best describes the graph? (A) y = 3cos(5 x ) + 4 (B) y = 3cos(4 x ) + 5 (C) y = 4cos(3 x ) + 5 (D) y = 5cos(3 x ) + 4 (E) y = 5sin(4 x ) + 3 Sample Problem -2p -1p 1p 2p 8 6 4 2

28. Find the baseline Vertical translation + 4 Find the height of peak Amplitude = 5 Number of waves in 2  Frequency = 3 Sample Problem -2p -1p 1p 2p 8 6 4 2 y = 5 cos( 3 x ) + 4

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