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Sin and Cos Graphs
Amplitude and Period
Objective: The student, given one of the six trigonometric
functions in standard form [e.g., y = A sin (Bx + C) + D, where
A, B, C, and D are real numbers], will: a) state the domain and
the range of the function; b) determine the amplitude, period,
phase shift, and vertical shift; and c) sketch the graph of the
function by using transformations for at least a one-period
interval.
Sin and Cos Graphs
•To graph the equation y = sin x or y = cos x, begin by
making a table of values of x and y that satisfy the equation:
x
0
y = sin x
0

4
2
2

6
1
2
Decimal
Approx.

3
3
2
0.7 0.9

2
1
2 3 5 
3
4
6
3
2 1
2
2
2

0
7  5  4  3 5  7  11
2
3
6
6
4
3
4
2
1
2
3
3
2 1
 

-1 
0


2
2
2
2
2
2
-0.7 -0.9
0.9 0.7
-0.9 -0.7
y = sin x
1
x
-1

6

4

3

2
2 3 5 
3 4 6

7  5 4 
6 4 3
3
2
5  7  11
3 4 6
2
Sin and Cos Graphs
•To graph the equation y = sin x or y = cos x, begin by
making a table of values of x and y that satisfy the equation:
x
0
y = cos x
1
Decimal
Approx.

4
2
2

6
3
2
2 3 5 
7  5  4  3

3
4
3
6
6
4
2
1
2
3
3
2 1

0
0  
-1 


2
2
2
2
2
2

3
1
2

2
-0.7 -0.9
0.9 0.7
5
3
1
2
-0.9 -0.7
7  11
2
6
4
2 3
1
2 2
0.7 0.9
y = cos x
1
x
-1

6

4

3

2
2 3 5 
3 4 6

7  5 4 
6 4 3
3
2
5  7  11
3 4 6
2
Sin and Cos Graphs
• The sine and cosine functions are
periodic functions, due to the nature of
the values periodically repeating
themselves.
Amplitude and Period
• Period:
– The period of a function f(x) is the smallest
positive number, p, for which f(x + p) = f(x) for all
x. The period is the length of the graph before it
begins to repeat itself.
– For the sine and cosine functions, the period is 2
since it is the smallest positive number for which
the values begin to repeat themselves.
• Amplitude:
– If the greatest value of y is M and the least value
of y is m, then the amplitude of the graph of y is
defined to be A = ½ |M – m|. The amplitude
determines the height of the graph.
5 Point Method of Graphing:
• When graphing the sine and cosine
functions, it is not necessary to graph
every point along the way. There are
FIVE critical points for each function.
• The five points occur at the beginning,
and end of the period, halfway through
the period, and at the ¼ and ¾ points
of the period.
Critical Points for Sine:
•
•
•
•
•
Beginning: (0, 0)
¼ : (/2, 1)
Halfway: (, 0)
y = sin x
¾ : (3/2, -1)
End: (2, 0)
1
x
/
-1
2

3/
2
2
Critical Points for Cosine:
•
•
•
•
•
Beginning: (0, 1)
¼ : (/2, 0)
Halfway: (, -1)
y = cos x
¾ : (3/2, 0)
End: (2, 1)
1
x
/
-1
2

3/
2
2
Sine and Cosine Graphs
• The graphs of y = sin x and y = cos x
both had a period of 2 and an
amplitude of 1.
• Both the period and amplitude of the
graphs of these trig functions can be
changed to produce variations of these
graphs.
• In general the graph of y = a sin bx and
y = a cos bx have an amplitude of |a|
and a period of 2/b.
Examples:
• State the amplitude and period for each
of the following trig functions.
a
b
1. y = 2 cos 2x
a
b
2. y = -3 sin 4x
Amp = |2| = 2
Amp = |-3| = 3
Per =
Per =
2/
2
=
2/
4
= /2
a
b
3. y = ½ sin 2/3 x
Amp = |½| = ½
Per =
2/
1
• 3/2 = 3
Sine and Cosine Graphs:
• We can graph these functions the same
way that we graph the regular
functions y = sin x and y = cos x, using
the 5 point method.
y = sin x
baseline
maximum
baseline
minimum
baseline
y = cos x
maximum
baseline
minimum
baseline
maximum
Sine and Cosine Graphs:
• To graph each of these functions, you
should:
– Find the amplitude and period.
– Divide the period into fourths and label the
x-axis.
– Use the amplitude as the maximum and
minimum to label the y-axis.
– Plot the 5 critical points and connect them
with a smooth curve.
Examples:
• Determine the period and amplitude of each
function. Then graph the function.
– y = 3 sin 2x
• Period: 
• Amp: 3
y
3
2
1
Domain:
−∞, ∞
Range: [-3,3]
/
-1
-2
-3
4
/
2
3/
4

x
Examples:
• Determine the period and amplitude of each
function. Then graph the function.
– y = 2 cos 0.5x
• Period: 4
• Amp: 2
y
3
2
1
Domain:
−∞, ∞
Range: [-2 ,2]

-1
-2
-3
2
3
4
x
Examples:
• Determine the period and amplitude of each
function. Then graph the function.
– y = -0.5 sin 3x
2/
• Period:
• Amp: 0.5
y
3
3
2
1
Domain:
−∞, ∞
Range: [-0.5 0.5]
/
-1
-2
-3
6
/
3
/
2
2/
x
3
Examples:
• Determine the period and amplitude of each
function. Then graph the function.
– y = 1.5 cos 2/3 x
• Period: 3
• Amp: 1.5
y
3
2
1
Domain:
−∞, ∞
Range: [-1.5, 1.5]
3/
-1
-2
-3
4
3/
2
9/
4
3
x