4.4 graphs of sine and cosine: sinusoids
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4.4 Graphs of Sine and Cosine: Sinusoids. By the end of today, you should be able to:. Graph the sine and cosine functions Find the amplitude, period, and frequency of a function Model Periodic behavior with sinusoids. Unit Circle. The Sine Function: y = sin(x ). Domain: Range: - PowerPoint PPT PresentationTRANSCRIPT
4.4 Graphs of Sine and Cosine: Sinusoids
By the end of today, you should be able to:
•Graph the sine and cosine functions•Find the amplitude, period, and frequency
of a function•Model Periodic behavior with sinusoids
Unit Circle
The Sine Function: y = sin(x)
•Domain:
•Range:
•Continuity:
•Increasing/Decreasing:
•Symmetry:
•Boundedness:
•Absolute Maximum:
•Absolute Minimum:
•Asymptotes:
•End Behavior:
The Cosine Function: y = cos(x)
•Domain:
•Range:
•Continuity:
•Increasing/Decreasing:
•Symmetry:
•Boundedness:
•Maximum:
•Minimum:
•Asymptotes:
•End Behavior:
Any transformation of a sine function is a Sinusoid
f(x) = a sin (bx + c) + d
Any transformation of a cosine function is also a sinusoid
•Horizontal stretches and shrinks affect the period and frequency
•Vertical stretches and shrinks affect the amplitude •Horizontal translations bring about phase shifts
The amplitude of the sinusoid:
f(x) = a sin (bx + c) +d
or
f(x) = a cos (bx+c) + d
is:
The amplitude is half the height of the wave.
a
Find the amplitude of each function and use the language of transformations to describe how the graphs are related to y = sin xy = 2 sin x
y = -4 sin x
You Try!
y = 0.73 sin x
y = -3 cos x
The period (length of one full cycle of the wave) of the sinusoid
f(x) = a sin (bx + c) + d and f(x) = a cos (bx + c) + d
is: 2πb
When : horizontal stretch by a factor of b <1 1b
If b < 0, then there is also a reflection across the y-axis
b >1When : horizontal shrink by a factor of 1b
Find the period of each function and use the language of transformations to describe how the graphs are related to y = cos x.
y = cos 3x
y = -2 sin (x/3)
You Try!y = cos (-7x)
y = 3 cos 2x
The frequency (number of complete cycles the wave completes in a unit interval) of the sinusoid
f(x) = a sin (bx + c) + d
and f(x) = a cos (bx + c) + d
is: b2π
Note: The frequency is simply the reciprocal of the period.
Find the amplitude, period, and frequency of the function:
You Try!
y=−32sin2x
y=2cosx3
Identify the maximum and minimum values and the zeros of the function in the interval
y = 2 sin x
−2π ,2π[ ]
y=3cosx2
Ex) Write the cosine function as a phase shift of the sine function
Ex) Write the sine function as a phase shift of the cosine function
Getting one sinusoid from another by a phase shift
Combining a phase shift with a period change
Construct a sinusoid with period and
amplitude 6 that goes through (2,0)
π5
Select the pair of functions that have identical graphs:y=cosx
y=sin x+π2
⎛⎝⎜
⎞⎠⎟
y=cos x+π2
⎛⎝⎜
⎞⎠⎟
Select the pair of functions that have identical graphs:
y=sin x+π2
⎛⎝⎜
⎞⎠⎟
y=−cos x−π( )
y=cos x−π2
⎛⎝⎜
⎞⎠⎟
HomeworkPg. 394-395
4, 12, 16, 20, 28, 33, 37, 38, 48, 54, 56, 58, 64
4.5 - Graphs of Tangent, Cotangent, Secant, and Cosecant
y = tan x
•Domain:
•Range:
•Continuity:
•Increasing/Decreasing:
•Symmetry:
•Boundedness:
•Asymptotes:
•End Behavior:
•Period
tan x =sinxcosx
Asymptotes at the zeros of cosine because if the denominator (cosine) is zero, then the function (tangent x) is not defined there.
Zeros of function (tan x) are the same as the zeros of sin (x) because if the numerator (sin x) is zero, then it makes the who function (tan x) equal to zero.
y = cot x
•Domain:
•Range:
•Continuity:
•Increasing/Decreasing:
•Symmetry:
•Boundedness:
•Asymptotes:
•End Behavior:
•Period
cot x =cosxsinx
Secant Functiony = sec x
•Domain:
•Range:
•Continuity:
•Increasing/Decreasing:
•Symmetry:
•Boundedness:
•Asymptotes:
•End Behavior:
•Period:
Cosecant Functiony = csc x
•Domain:
•Range:
•Continuity:
•Increasing/Decreasing:
•Symmetry:
•Boundedness:
•Asymptotes:
•End Behavior:
•Period: