1. Graphs of Trigonometric Functions
Digital Lesson
Graphs of Trigonometric Functions

2. 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.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Properties of Sine and Cosine Functions

3. 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
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Sine Function

4. 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 x
y = cos x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Cosine Function

5. 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)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: y = 3 cos x

6. 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 = sin x
y = sin x
y = 2 sin x
y = – 4 sin x
reflection of y = 4 sin x
y = 4 sin x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Amplitude

7. 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 0 < b < 1, the graph of the function is stretched horizontally. y x
period:
period: 2
If b > 1, the graph of the function is shrunk horizontally. y x
period: 4
period: 2
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Period of a Function

8. 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)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Graph y = f(-x)

9. Use the identity sin (– x) = – sin x:
Example: 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)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: y = 2 sin(-3x)

10. 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
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

11. Example
Determine the amplitude, period, and phase shift of y = 2sin(3x-)
Solution:
Amplitude = |A| = 2
period = 2/B = 2/3
phase shift = C/B = /3
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

12. Example cont. y = 2sin(3x- )
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

13. Amplitude Period: 2π/b Phase Shift: c/b Vertical Shift
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.