Download Graphs of Trigonometric Functions

MATH 130
Lecture on
The Graphs of
Trigonometric Functions
1
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  1  y  1.
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 2 .
6. The cycle repeats itself indefinitely in both directions of the
x-axis.
2
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.

3
x
0

2
2
sin x
0
2
1
0
-1
0
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 = sin x
y
3

2



1

2
2

3
2
2
5
2
x
1
3
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.
x
y = 3 cos x
y

(0, 3)
2
1

0
3
0

-3
x-int
min
2
max
3
2
0
2
3
x-int
max
(2, 3)

1 
( , 0)
2
2
3
( , –3)
2
( 3 , 0)
2
3
4 x
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.

3
x
0

2
2
cos x
1
2
0
-1
0
1
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
3

2



1

2
2

3
2
2
5
2
x
1
5
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
4
y = 2sin x

2
y=
1
2

3
2
2
x
sin x
y = – 4 sin x
reflection of y = 4 sin x
y = sin x
y = 4 sin x
4
6
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 2 .
b
For b  0, the period of y = a cos bx is also 2 .
b
If 0 < b < 1, the graph of the function is stretched horizontally.
y
period : 
period: 2
y  sin 2 x
y  sin x x


2
If b > 1, the graph of the function is shrunk horizontally.
y
y  cos x
1
y  cos x
period: 2
2 
2
3
4

x
period: 4
7
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 = sin (–x)
y
Use the identity
sin (–x) = – sin x
y = sin x
x

2
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
Use the identity
x
cos (–x) = – cos x

2
y = cos (–x)
8
Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin bx with b > 0
y = 2 sin (–3x) = –2 sin 3x
Use the identity sin (– x) = – sin x:
2  2
period:
amplitude: |a| = |–2| = 2
=
3
b
Calculate the five key points.
x
0
y = –2 sin 3x
0
y



6
3
2
2
3
–2
0
2
0
(  , 2)
2
6


6
3
(0, 0)
2

(  ,-2)
2
2
3

2
5
6

x
(  , 0) 2
3
( , 0)
3
6
9
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.
y
amplitude = | A|
period = 2 /B.
y = A sin Bx
Amplitude: | A|
x
Starting point: x = C/B
Period: 2/B
10
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
11
Example cont.
• y = 2sin(3x- )
3
2
1
-6 -5 -4 -3 -2 -1
1 2 3 4 5 6
-1
-2
-3
12
A common mistake…
a is not amplitude; a is amplitude.
a may be positive or negative; amplitude is always
positive.
The standard forms for sine and cosine functions are:
f (t )  a sin(bt  c)  d
g (t )  a cos(bt  c)  d
where a,b,c and d are constants
.
13
In the standard form:
f (t )  a sin(bt  c)  d
g (t )  a cos(bt  c)  d
•a controls amplitude
•b controls period
•c controls phase shift
•d controls vertical shift
14
a sin( bx  c)  d
Amplitude
Period:
2π/b
Phase Shift:
c/b
Vertical
Shift
15