14.1 Graph Sine, Cosine, and
Tangent Functions
VOCABULARY
Amplitude
The
amplitude of the graph of a sine or cosine function is half the difference of
the maximum M and the minimum m, or (M - m).
Periodic function
A
function whose graph has a repeating pattern
Cycle
The repeating pattern of a periodic function
Period
The horizontal length of a cycle
Frequency
The reciprocal of the period; the number of cycles
per unit of time
CHARACTERISTICS OF y
= sin x AND y = cos x
1. The domain of
each function is _all real numbers_.
2. The _range_
of each function is -1 £ y £ 1. Therefore, the minimum
value of each function is m = -1 and the maximum value is M
= 1.
3.
The _amplitude_
of each function's graph is half the difference of the maximum M and the
minimum m, or (M - m) = [(1 - (-1)] = 1.
4.
Each function is periodic, which means that its
graph has a _repeating_ pattern, called a cycle. The horizontal length
of each cycle is called the _period_ .
5.
The x-intercepts of y = _sin x_
occur when x = 0, ± p, ± 2p, ± 3p .....
6.
The x-intercepts of y = _cos x_
occur when x
AMPLITUDE AND PERIOD
The
amplitude and period of the graphs of y = a sin bx and y
= a cos bx, where a and b are
nonzero real numbers, are:
_Amplitude_ =
½a½ _Period_
=
Example
1
Graph
sine and cosine functions
Graph
(a) y = 2 sin x and (b) y = cos px.
Solution
a.
Notice
how change in a and b affect the
graphs of y = a sin bx and y = a cos bx.
When the value of a increase, the amplitude
increases. When the value of b increases, the period decreases.
The amplitude is a = _2_
and the period is .
Intercepts: (0, 0);
Maximum:
Minimum:
b.
The amplitude is a = and the period is .
Intercepts:
Maximums:
Minimum:
Graph the
function.
1. 
y = sin 2px
2. y = 3 cos x
Example
2
Model with a sine
function
Write
a sine function with an amplitude of 3 and a frequency
of 1000.
Solution
Find the values of a and
b in the equation y = a sin bx.
The
amplitude is 3, so a = 3. Use the frequency to find b. The
frequency is the reciprocal of the period. So, b = _2000p_
. The
equation is _y = 3 sin 200077px_ .
Write a sine function with
the given amplitude and frequency.
3. amplitude = 4
frequency = 1500
y = 4 sin 3000px
4. amplitude = 1.5
frequency = 500
y = 1.5 sin 1000px
CHARACTERISTICS
OF y = a tan bx
The period and vertical asymptotes of the graph of y
= a tan bx, where a and b
are nonzero real numbers, are:
The period is 
The vertical asymptotes are at odd multiples of 
.
Example
3
Graph
a tangent function
Graph one period of the
function y = 2 tan x.
Solution
The period is .
Intercept:
_(0, 0)_
Asymptotes: x =
Halfway points:
Graph the
function.
5. y = tan 4x
6.
y = tan px