graphing sine and cosine functions

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


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.


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


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


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


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=


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

–


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

, ,

graphing sine and cosine functions

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