1. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 5
Trigonometric Functions
5.5 Part 2 Graphs of Sine and Cosine Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

2. Objectives:
Understand the graph of y = sin x.
Understand the graph of y = cos x.
Graph variations of y = sin x.
Graph variations of y = cos x.
Use vertical shifts of sine and cosine curves.
Model periodic behavior.

3. Period
Because y = a sin x completes one cycle from x = 0 to x = 2, it follows that y = a sin bx completes one cycle from x = 0 to x = 2 /b, where b is a positive real number.
For b > 0, the graph of y = sin bx will resemble that of y = sin x, but with period 2/b. The graph of y = cos bx will resemble that of y = cos x, with period 2/b.

4. Period
Note; when b > 1, the period of y = a sin bx (or y = a cos bx) is less than 2 and represents a horizontal shrinking (smaller period) of the graph of y = a sin x (or y = a cos bx). when 0 < b < 1, the period of y = a sin bx (or y = a cos bx) is greater than 2 and represents a horizontal stretching (larger period) of the graph of y = a sin x (or y = a cos bx).

5. Period
When b is negative, the identities sin(–x) = –sin x and cos(–x) = cos x are used to rewrite the function and make b positive.

6. Period
The period is the distance between two peaks or valleys.
y = sin x has period 2π
y = sin(bx) has period 2π/b

7. Period The length of a Period (or simply Period)
= (horizontal distance between adjacent peaks)
= (horizontal distance between adjacent troughs)
one period

8. If b > 1, the graph of the function is shrunk horizontally.
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 2𝜋 𝑏 .
For b  0, the period of y = a cos bx is also 2𝜋 𝑏 .
If b > 1, the graph of the function is shrunk horizontally. y x
period: π
period: 2π

9. Period
If 0 < b < 1, the graph of the function is stretched horizontally. y x
period: 4π
period: 2π

10. Example: Graph y = cos 2x/3 over one period
For y = cos 2x/3, b = 2/3. Then, the period (2π/b = 2π/(2/3) = 3π) is 3. The graph will complete one period over the interval [0, 3].
The endpoints are 0 and 3, the three middle points are:
1 4 3𝜋 , 𝜋 , 𝜋
Make a table of the 5 key points, plot the points and join in a smooth curve.

11. Example: Graph y = cos 2x/3 over one period
The period is 3.
Divide the interval into four equal parts.
Obtain key points for one period. 1 1
cos 2x/3 2
3/2 
/2
2x/3 3
9/4
3/4 x

12. Example: Graph y = cos 2x/3 over one period continued
The amplitude is 1.
Join the points and connect with a smooth curve.

13. Your Turn: Graph y = sin 2x over one period

14. Graphing Variations of y = a sin bx and y = a cos bx
Identify the amplitude (|a|) and the period (2π/b). Start with 0 on the x-axis, and lay off a distance of one period (2/b).
Divide the interval on the x-axis into four equal parts. Find the x values for the 5 key points; the x-intercepts, maximum and minimum points. Start with the value of x where the period begins and add quarter-periods (that is, 𝑝𝑒𝑟𝑖𝑜𝑑 4 ) to find successive values of x.
Find the values of y by evaluating the function for each of the five x-values resulting from Step 2. Make a table of values.
Connect the 5 key points with a smooth curve and graph one complete period of the given function.
Extend the graph in step 4 to the left or right as needed.

15. Example #1: Graph y = 2 sin 4x
Amplitude: a = -2 → |-2| = 2 amplitude, a = -2 → negative = reflected over x-axis. Period: b = 4 → 2π/b = 2/4 = /2 period. The function will be graphed over the interval [0, /2].
Divide the interval into four equal parts: 0, ( 1 4 ∙ 𝜋 2 ), ( 1 2 ∙ 𝜋 2 ), ( 3 4 ∙ 𝜋 2 ), π/2 → 0, 𝜋 8 , 𝜋 4 , 3𝜋 8 , 𝜋 2 .

16. Example #1: (continued) y = 2 sin 4x
Make a table of values 2 2
2 sin 4x 1 1
sin 4x 2
3/2 
/2 4x
3/8
/4
/8 x

17. Example #1: (continued) y = 2 sin 4x
Plot the 5 key points and join them with a sinusoidal curve with amplitude 2.
/4
3/8
/2
/8
/2
/8 x y
/4
3/8 -2 2
Extend the graph as needed.

18. Your Turn #1:
Determine the amplitude and period of 𝑦=2 sin 1 2 𝑥 . Then graph the function for 0 ≤ x ≤ 8π.

19. Find the amplitude and the period.
Example #2:
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EXAMPLE 6
Graphing y = A cos (Bx)
Graph 𝑦=3 cos 𝑥
Solution
Find the amplitude and the period.

20. Example #2: (continued) 𝑦=3 cos 1 2 𝑥
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Example #2: (continued) 𝑦=3 cos 𝑥
2. Find the x-coordinates for the five key points.
1 4 (period) = (4π) = π
x1 = 0
x2 = ∙ 4π = π
x3 = 1 2 ∙ 4π = 2π
x4 = 3 4 ∙ 4π = 3π
x5 = 4π

21. Example #2: (continued) 𝑦=3 cos 1 2 𝑥
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Example #2: (continued) 𝑦=3 cos 𝑥
3. Find the y-coordinates for the five key points.

22. Example #2: (continued) 𝑦=3 cos 1 2 𝑥
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Example #2: (continued) 𝑦=3 cos 𝑥
4. Graph of 𝑦=3 cos 𝑥 , 0 ≤ x ≤ 4π

23. Example #2: (continued) 𝑦=3 cos 1 2 𝑥
© 2010 Pearson Education, Inc. All rights reserved
Example #2: (continued) 𝑦=3 cos 𝑥
5. Extend Graph of 𝑦=3 cos 𝑥 , as needed.

24. Graph y = –3 cos x over one period.
Your Turn #2:
GRAPHING y = a cos bx FOR b THAT IS A MULTIPLE OF π
Graph y = –3 cos x over one period.

25. Graph y = –2 sin 3x over one period.
Your Turn #3:
GRAPHING y = a sin bx
Graph y = –2 sin 3x over one period.