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 1

b

If b < 0, then there is also a reflection across the y-axis

b >1When : horizontal shrink by a factor of 1

b

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: b

2π

Note: The frequency is simply the reciprocal of the period.

Find the amplitude, period, and frequency of the function:

You Try!

y=−32

sin2x

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

⎛⎝⎜

⎞⎠⎟

Homework

Pg. 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: