14.1 Graphing Sine, 14.1 Graphing Sine, Cosine and Tangent Cosine and Tangent FunctionsFunctionsAlgebra 2

The graph of y=sin xThe graph of y=sin x

The graph of y=cos xThe graph of y=cos x

Characteristics of y=sinx and Characteristics of y=sinx and y=cosxy=cosxThe domain is all real numbersThe range is -1 ≤ y ≤ 1The function is periodic-the graph

has a repeating pattern. (Shortest repeating pattern called a cycle and the horizontal length is called the period) Both have a period of 2π.

Characteristics of y=sinx and Characteristics of y=sinx and y=cosxy=cosxThe maximum value of y=sinx is

M=1 and occurs when The maximum value of y=cosx is

M=1 and occurs when x=2nπ.The minimum value of y=sinx is

m=-1 and occurs at The minimum value of y=cosx is

m=-1 and occurs when x=(2n+1)π

nx 2

2

nx 2

2

3

Characteristics of y=sinx and Characteristics of y=sinx and y=cosxy=cosxThe amplitude of both functions

is

Amplitude is half the height of the graph.

12

1 mM

Characteristics of y=a sin bx Characteristics of y=a sin bx and y=a cos bxand y=a cos bxThe amplitude and period of the

graphs of y = a sin bx and y = a cos bx where a and b are nonzero numbers are as follows.

amplitude=

period=

a

b

2

Examples:Examples:Graph the functions

◦

◦

◦

xy cos2

1

xy2

1sin

4sin2x

y

Examples:Examples:Give the amplitude, period. And

five key points of the graph of each function.

◦

◦

◦

xy sin

xy cos3

4sin2x

y

DefinitionDefinitionFrequency- the number of cycles

per unit of time (frequency is the reciprocal of the period)

Examples:Examples:A tuning fork vibrates with

frequency f=880 hertz (cycles per second.) You strike the tuning fork with a force that produces a maximum pressure of 4 pascals.◦Write a sine model that gives the

pressure P as a function of t (in seconds).

◦Graph the model.

Examples:Examples:You pluck the string of a violin so

that it vibrates with frequency f = 660 hertz (cycles per second.) The force of the pluck produces a maximum pressure of 2 pascals. Write a sine model that gives the pressure P as a functions of time t (in seconds). Then give the amplitude and period of the function's graph.

Tangent FunctionsTangent FunctionsThe graph of y=tanx has the

following characteristics.◦The domain is all real numbers

except odd multiples of . At , the graph has vertical asymptotes

◦The range is all real numbers.◦The graph has a period of π

2

2

Characteristics of y=a tan Characteristics of y=a tan bxbxIf a and b are nonzero real

numbers, the graph of y= a tan bx has these characteristics.◦The period is

◦There are vertical asymptotes at odd multiples of

b

b2

Examples:Examples:Graph the functions.

◦

◦

xy2

tan3

xy3

1tan2

1

QuestionQuestionHow do you find the amplitude,

period and vertical asymptotes of a sine, cosine, or tangent function from its equation?