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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?