graphing sine and cosine functions in this lesson you will learn to graph functions of the form y =...
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Graphing Sine and Cosine Functions
In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are positive constants and x is in radian measure. The graphs of all sine and cosine functions are related to the graphs of y = sin x and y = cos xwhich are shown below.
y = sin x
y = cos x
Graphing Sine and Cosine Functions
The domain of each function is all real numbers.
The range of each function is –1 y 1.
Each function is periodic, which means that its graph has a repeating pattern that continues indefinitely.
The maximum value of y = sin x is M = 1 and occurs when x = + 2 n π where n is any integer. The maximum value of y = cos x is also M = 1 and occurs when x = 2 n π where n is any integer.
π2
The functions y = sin x and y = cos x have the following characteristics:
The horizontal length of each cycle is called the period.
The shortest repeating portion is called a cycle.
The graphs of y = sin x and y = cos x each has a period of 2π.
Graphing Sine and Cosine Functions
The functions y = sin x and y = cos x have the following characteristics:
The amplitude of each function’s graph is (M – m) = 1.12
The minimum value of y = sin x is m = –1 and occurs when x = + 2 n π where n is any integer. The minimum value of y = cos x is also m = –1 and occurs when x = (2n + 1)π where n is any integer.
3π2
CHARACTERISTICS OF Y = A SIN BX AND Y = A COS BX
Graphing Sine and Cosine Functions
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 as follows:
amplitude = |a| and period = 2π|b|
The graph of y = 2 sin 4x has amplitude 2 and period = .2π4
π2
The graph of y = cos 2 π x has amplitude and period = 1.2π2π
13
13
Examples
CHARACTERISTICS OF Y = A SIN BX AND Y = A COS BX
Graphing Sine and Cosine Functions
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 as follows:
amplitude = |a| and period = 2π|b|
The graph of y = 2 sin 4x has amplitude 2 and period = .2π4
π2
The graph of y = cos 2 π x has amplitude and period = 1.2π2π
13
13
Examples
For a > 0 and b > 0, the graphs of y = a sin bx and y = a cos bx each have five key x-values on the interval 0 x : the x-values at which the maximum and minimum values occur and the x-intercepts.
2πb
Graphing Sine and Cosine Functions
Graph the function.
y = 2 sin x
SOLUTION
The amplitude is a = 2 and the period is = = 2π. The five key points are:2πb
2π1
Intercepts: (0, 0); (2π, 0); = (π, 0)( )• 2, 012
Minimum: ( )• 2, –234
Maximum: ( )• 2, 214 ( ), 2
2=
( ), –232=
Graphing Sine and Cosine Functions
Graph the function.
SOLUTION
Maximums: (0, 1); (π, 1)
The amplitude is a = 1 and the period is = = π. The five key points are:2πb
2π2
y = cos 2x
Intercepts: = ; = , 0( )4( )• , 01
4 ( )• , 034 ( ), 03
4
Minimum: =( )• , –112 ( ), –1
2
Graphing a Cosine Function
SOLUTION
Graph y = cos π x.13
The five key points are:
The amplitude is a = and the period is = = 2. 13
2πb
2ππ
Intercepts: = ; =( )• 2, 014 ( ), 0
12 ( )• 2, 0
34 ( ), 0
32
Maximum: ;( )0,13 ( )2,
13
Minimum: =( )• 2,12
13
– ( )1,13
–
Graphing Sine and Cosine Functions
The periodic nature of trigonometric functions is useful for modeling oscillating motions or repeating patterns that occur in real life.
In such applications, the reciprocal of the period is called the frequency.
Some examples are sound waves, the motion of a pendulum or a spring, and seasons of the year.
The frequency gives the number of cycles per unit of time.
Modeling with a Sine Function
MUSIC When you strike a tuning fork, the vibrations cause changes in the pressure of the surrounding air. A middle-A tuning fork vibrates with frequency f = 440 hertz (cycles per second). You would strike a middle-A tuning fork with a force that produces a maximum pressure of 5 pascals.
SOLUTION
Write a sine model that gives the pressure P as a function of time t (in seconds).Then graph the model.
In the model P = a sin b t, the maximum pressure P is 5, so a = 5. You can use the frequency to find the value of b.
Frequency = 1period
440 =b
2π
880π = b
The pressure as a function of time is given by P = 5 sin 880π t.
Modeling with a Sine Function
MUSIC When you strike a tuning fork, the vibrations cause changes in the pressure of the surrounding air. A middle-A tuning fork vibrates with frequency f = 440 hertz (cycles per second). You would strike a middle-A tuning fork with a force that produces a maximum pressure of 5 pascals.
The five key points are:
The amplitude is a = 5 and the period is = .1
4401f
SOLUTION
Write a sine model that gives the pressure P as a function of time t (in seconds).Then graph the model.
Intercepts: (0, 0); ; =( )1440
, 0 ( )• , 012
1440 ( )1
880, 0
Maximum: =( )• , 514
1440 ( )1
1760, 5
Minimum: = ( )31760
, –5( )• , –534
1440
Graphing Tangent Functions
The domain is all real numbers except odd multiples of . At odd multiples of , the graph has vertical asymptotes.
π2
π2
The range is all real numbers. The graph has a period of π.
If a and b are nonzero real numbers, the graph of y = a tan bx has these characteristics:
The period is . π | b |
There are vertical asymptotes at odd multiples of . π
2| b |
CHARACTERISTICS OF Y = A TAN BX
The graph of y = tan x has the following characteristics.
Example The graph of y = 5 tan 3x has period and asymptotes at
x = (2n + 1) = + where n is any integer.
π 3
π 6
n π 3
π 2(3)
Graphing Tangent Functions
The graph at the right shows five key x-values that can help you sketch the graph of y = tan x for a > 0 and b > 0.
The graph of y = tan x has the following characteristics.
These are the x-intercept, the x-values where the asymptotes occur, and the x-values halfway between the x-intercept and the asymptotes.
At each halfway point, the function’s value is either a or – a.
Graphing a Tangent Function
SOLUTION
The period is = .πb
π4
Intercept: (0, 0)
x = • , or x = ;π8
12
π4
x = – • , or x = – π8
12
π4
Halfway points:
Asymptotes:
Graph the function y = tan 4x.32
,( ) ( );• =14
π4
32
π16
32
,
( ) ( )– • – = – –14
32
π4
π16
32
, ,