1. Graphs of Sine and Cosine Functions
4.5

2. Objectives Sketch the graphs of basic sine and cosine functions.
Use amplitude and period to help sketch the graphs of sine and cosine functions.
Sketch translations of the graphs of sine and cosine functions.
Use sine and cosine functions to model real-life data.

3. Periodic Functions
Many things in daily life repeat with a predictable pattern, such has weather, tides, and hours of daylight.
If a function repeats its values in a regular pattern, it is called a periodic function.
The following is a periodic graph that represents a normal heartbeat.

4. Periodic Function Definition
A function f is called a periodic function if there is a positive real number p such that
f (x) = f (x + np)
for every real number x in the domain of f and every integer n.
The least possible value of p is called the period of the function.

5. The Circular Functions
The trigonometric functions, when defined for all real values, are referred to as the circular functions. To define the circular functions for any real number s, use the unit circle, the circle with center at the origin and radius one unit. Start at the point (1,0) and measure an arc of length s, counterclockwise if s > 0 and clockwise if s < 0.

6. Basic Sine and Cosine Curves

7. The Circular Functions
sinθ = y
cosθ = x

8. Basic Sine and Cosine Curves
In this section, you will study techniques for sketching the graphs of the sine and cosine functions. The graph of the sine function is a sine curve.

9. 315°
225°
270° 0°
180°
90°
45°
30°
330°
300°
60°
240°
135°
210°
150°
120°
y = sin x 1
90°
180°
270°
360° –1

10. Basic Sine and Cosine Curves
In Figure 1.36, the black portion of the graph represents one period of the function and is called one cycle of the sine curve. The gray portion of the graph indicates that the basic sine curve repeats indefinitely to the left and right.
Figure 1.36

11. 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. -1 1
sin x x
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

12. Sine Curve
For this cycle, notice these 5 key points: the 3 intercepts and the min/max points. These are all you really need to plot your sine curve.
Sine = Snake

13. Sine Curve Stuff Sine = Snake
Domain: All Real #s Range: [−1, 1] Period: 2π Looks Like: A Snake Zeros: {…, −2π, −π, 0, π, 2π, …} Symmetry: Origin

14. Basic Sine and Cosine Curves
The graph of the cosine function is shown in Figure 1.37.
Figure 1.37

15. 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. 1 -1
cos x x
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 = cos x y x

16. Cosine Curve
For this cycle, notice these 5 key points: the 3 intercepts and the min/max points. These are all you really need to plot your cosine curve.
Cosine = Cup

17. Cosine Curve Stuff Cosine = Cup
Domain: All Real #s Range: [−1, 1] Period: 2π Looks Like: A Cup Zeros: {…, −3π/2, −π/2, π/2, 3π/2, …} Symmetry: y-axis
Cosine = Cup

18. Comparing Sin & Cos Functions

19. Basic Sine and Cosine Curves
We know that the domain of the sine and cosine functions is the set of all real numbers. Moreover, the range of each function is the interval [–1, 1], and each function has a period of 2. Do you see how this information is consistent with the basic graphs shown in Figures 1.36 and 1.37?
Figure 1.36
Figure 1.37

20. Basic Sine and Cosine Curves
Note in Figures 1.36 and 1.37 that the sine curve is symmetric with respect to the origin, whereas the cosine curve is symmetric with respect to the y-axis. These properties of symmetry follow from the fact that the sine function is odd and the cosine function is even.

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

22. Basic Sine and Cosine Curves
To sketch the graphs of the basic sine and cosine functions by hand, it helps to note five key points in one period of each graph: the intercepts, maximum points, and minimum points (see below).

23. Example – Using Key Points to Sketch a Sine Curve
Sketch the graph of y = 2 sin x on the interval [–, 4 ]. Solution: Note that y = 2 sin x indicates that the y-values for the key points will have twice the magnitude of those on the graph of y = sin x. Divide the period 2 into four equal parts to get the key points Intercept Maximum Intercept Minimum Intercept and .
= 2(sin x)

24. Example – Solution
cont’d
By connecting these key points with a smooth curve and extending the curve in both directions over the interval [–, 4 ], you obtain the graph shown below.

25. Transformations of Trigonometric Functions

26. Transformations Vertical Translation
Horizontal Translation (phase shift)
Dilations
Amplitude Change
Period Change
Reflection

27. Amplitude and Period

28. Amplitude and Period
In the rest of this section, you will study the graphic effect of each of the constants a, b, c, and d in equations of the forms y = d + a sin(bx – c) and y = d + a cos(bx – c). The constant factor a in y = a sin x acts as a scaling factor—a vertical stretch or vertical shrink of the basic sine curve. When | a | > 1, the basic sine curve is stretched, and when | a | < 1, the basic sine curve is shrunk.

29. Amplitude and Period
The result is that the graph of y = a sin x ranges between –a and a instead of between –1 and 1. The absolute value of a is the amplitude of the function y = a sin x. The range of the function y = a sin x for a  0 is –a  y  a.

30. Amplitude The amplitude is the maximum height above the center line.
The standard sine curve y = sin x has amplitude 1
y = A sin x has amplitude |A|

31. If |a| > 1, the amplitude stretches the graph vertically.
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 = 2 sin x
y = sin x
y = sin x
y = – 4 sin x
reflection of y = 4 sin x
y = 4 sin x

32. Amplitude Amplitude = (vertical distance between peak and trough)/2

33. Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.
max
x-int
min 3 -3
y = 3 cos x 2  x y x
(0, 3)
( , ,3)
( , 0)
( , 0)
( , –3)

34. Reflection
The graph of the function y = -a sin(bx) + d and y = -a cos(bx) + d is reflected about the x-axis compared to the graph y = a sin(bx) and y = a cos(bx ).
The negative of the function, y = -f(x), is a reflection about the x-axis.

35. Reflection
The graph of y = –f (x) is a reflection in the x-axis of the graph of y = f (x). For instance, the graph of y = –3 cos x is a reflection of the graph of y = 3 cos x, as shown in Figure 1.39.
Figure 1.39

36. Example: Sketch the graph of y = -sin (x).
Reflection
Example: Sketch the graph of y = -sin (x).
The graph of y = -sin (x) is the graph of y = sin x reflected in the x-axis.
y = -sin (x) y x
360˚
180˚
y = sin x

37. Example: Sketch the graph of y = -cos (x).
Reflection
Example: Sketch the graph of y = -cos (x).
The graph of y = -cos (x) is the graph of y = cos x reflected in the x-axis.
y = -cos (x) y x
360˚
180˚
y = cos x

38. Use basic trigonometric identities to graph y = f (–x)
Example 1: Sketch the graph of y = sin (–x).
The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. y x
y = sin (–x)
Use the identity sin (–x) = – sin x
y = sin x
Example 2: Sketch the graph of y = cos (–x).
The graph of y = cos (–x) is identical to the graph of y = cos x. y x
Use the identity cos (–x) = cos x
y = cos (–x)
y = cos (–x)

39. Period The period is the distance between two peaks or valleys.
y = sin x has period 2π
y = sin(Bx) has
period 2π/B

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

41. 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. Note that when 0 < b < 1, the period of y = a sin bx is greater than 2 and represents a horizontal stretching of the graph of y = a sin x.

42. Period
Similarly, when b > 1, the period of y = a sin bx is less than 2 and represents a horizontal shrinking of the graph of y = a sin x. When b is negative, the identities sin(–x) = –sin x and cos(–x) = cos x are used to rewrite the function.

43. If b > 1, the graph of the function is shrunk horizontally.
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 b > 1, the graph of the function is shrunk horizontally. y x
period:
period: 2
If 0 < b < 1, the graph of the function is stretched horizontally. y x
period: 4
period: 2

44. Example – Scaling: Horizontal Stretching
Sketch the graph of . Solution: The amplitude is 1. Moreover, because b = , the period is
Substitute for b.

45. Example – Solution
cont’d
Now, divide the period-interval [0, 4 ] into four equal parts with the values , 2, and 3 to obtain the key points Intercept Maximum Intercept Minimum Intercept (0, 0), (, 1), (2, 0), (3, –1), and (4, 0). The graph is shown below.

46. Use the identity sin (– x) = – sin x:
Your Turn: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin bx with b > 0
Use the identity sin (– x) = – sin x:
y = 2 sin (–3x) = –2 sin 3x
period: 2 3 =
amplitude: |a| = |–2| = 2
Calculate the five key points. 2 –2
y = –2 sin 3x x y x
( , 2)
(0, 0)
( , 0)
( , 0)
( , -2)

47. Translations of Sine and Cosine Curves

48. Translations of Sine and Cosine Curves
The constant c in the general equations y = a sin(bx – c) and y = a cos(bx – c) creates horizontal translations (shifts) of the basic sine and cosine curves. Comparing y = a sin bx with y = a sin(bx – c), you find that the graph of y = a sin(bx – c) completes one cycle from bx – c = 0 to bx – c = 2.

49. Translations of Sine and Cosine Curves
By solving for x, you can find the interval for one cycle to be This implies that the period of y = a sin(bx – c) is 2 /b, and the graph of y = a sin bx is shifted by an amount c/b. The number c/b is the phase shift.

50. Phase Shift The Graph of y = asin(bx - c)
The graph of y = a sin (bx – c) is obtained by horizontally shifting the graph of y = a sin bx so that the starting point of the cycle is shifted from x = 0 to x = c/b. The number c/b is called the phase shift.
amplitude = | a|
period = 2 /b. y
y = a sin bx
Amplitude: | a| x
Starting point: x = c/b
Period: 2/b

51. Phase Shift (horizontal shift)
is the amount of horizontal shift compare to a curve with the same period and passing through the origin (if it has no vertical shift). This value can be positive or negative.
A positive phase shift means the curve is shift to the right compared to the black curve.
A negative phase shift means the curve is shifted to the left compared to the black curve.
Amount of
positive shift
Amount of
negative shift
This black curve is for reference.

52. Translations of Sine and Cosine Curves

53. Example – Horizontal Translation
Sketch the graph of y = –3 cos(2 x + 4). Solution: The amplitude is 3 and the period is 2 / 2 = 1. By solving the equations 2 x + 4 = 0 2 x = –4 x = –2 and 2 x + 4 = 2

54. Example – Solution
cont’d
2 x = –2 x = –1 you see that the interval [–2, –1] corresponds to one cycle of the graph. Dividing this interval into four equal parts produces the key points Minimum Intercept Maximum Intercept Minimum and

55. Example – Solution The graph is shown in Figure 1.40 cont’d

56. Your Turn
Determine the amplitude, period, and phase shift of y = 2sin(3x-)
Solution:
y = 2sin(3x-π)
Amplitude = |a| = 2
period = 2/b = 2/3
phase shift = c/b = /3 (endpoints 𝜋 3 and 𝜋) or the interval is [𝜋/3, 𝜋]

57. Your Turn cont.
y = 2sin(3x- )

58. Vertical Translations
If y = f(x) is changed to y = f(x) + c, then the graph is shifted up by c units.
If y = f(x) is changed to y = f(x) - c, then the graph is shifted down by c units.

59. Translations of Sine and Cosine Curves
The final type of transformation is the vertical translation caused by the constant d in the equations y = d + a sin(bx – c) and y = d + a cos(bx – c). The shift is d units up for d > 0 and d units down for d < 0. In other words, the graph oscillates about the horizontal line y = d instead of about the x-axis.

60. Vertical shift: d y = a sin(bx – c) + d
is the amount that the magenta curve is moved up (or down) compared to the reference (black) curve.
A positive shift means an upward displacement while a negative shift means a downward displacement.
vertical shift
This black curve is for reference.

61. Vertical Translation - Example
y = (sin x) + 1
y = (sin x)
y = (sin x) – 1

62. Vertical Translation - Example
Vertical Translation = 3 units upward, since d > 0
- 2 2
60˚
30˚
90˚
120˚

63. Amplitude
Period: 2𝜋/b
Phase Shift: c/b
Vertical Shift

64. Conclusions Then |a| is the amplitude
Let y = a sin(bx – c) + d be the equation of the curve.
Then
|a| is the amplitude
b is inversely proportional to the period but directly proportional to frequency. In fact period = 2π / |b|
c is proportional to phase shift, more precisely phase shift = c / |b|
Left endpoint: bx – c = 0
Right endpoint: bx – c = 2𝜋
d is the vertical shift.

65. Example: Graphing y = a sin(bx – c) + d
Graph y = –1 + 2 sin(4x + ).
Solution Express y in the form a sin(bx – c)+d.
Amplitude = 2
Period
Translate |-1| = 1 unit
downward and units
to the left.
Start the first period at x-value and end the first period at

66. Example: Analyze y = 1 + 2 cos (4x) compared with y = cos x
Period: 360˚/4 = 90˚
Amplitude: 2
Vertical Translation: down 1 unit
Max. = = 1 & Min. = -1 – 2 = -3
Max: Vert. Translation + Amp.
Min: Vert. Translation – Amp.

67. Your Turn: Analyze and graph: Amp. = 2 Period = 360˚/3 = 120˚
Vertical translation = up 2 units
Reflection = (-2sin) about x-axis
Max. = = 4
Min. = 2 – 2 =0
30˚
120˚

68. Your Turn: Analyze and Graph y = 1 + 2 cos (4x)
Amp. = 2
Period 360˚/4 = 90˚
Vertical Translation = -1
Reflection = none
Max. = =1
Min. = -1 – 2 = -3
-45˚
90˚

69. Assignment:
Pg. 304 – 308:#3 – 33 odd, 39 – 45 odd, odd