4.4 Graphs of Sine and Cosine: Sinusoids - PowerPoint Presentation

Slide1

4.4 Graphs of Sine and Cosine: SinusoidsSlide2

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 sinusoidsSlide3

Unit CircleSlide4

The Sine Function: y =

sin(x

)

Domain:

Range:

Continuity:

Increasing/Decreasing:Symmetry:Boundedness:

Absolute Maximum:Absolute Minimum:Asymptotes:End Behavior:Slide5

The Cosine Function: y =

cos

(

x

)

Domain:

Range:

Continuity:

Increasing/Decreasing:

Symmetry:

Boundedness:

Maximum:Minimum:Asymptotes:End Behavior:Slide6

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 sinusoidHorizontal stretches and shrinks affect the period and frequencyVertical stretches and shrinks affect the amplitude

Horizontal translations bring about phase shiftsSlide7

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.Slide8

Find the amplitude of each function and use the language of transformations to describe how the graphs are related to y = sin

x

y = 2 sin

x

y = -4 sin

x

You Try! y = 0.73 sin x

y = -3 cos xSlide9

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:

When : horizontal stretch by a factor of

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

When : horizontal shrink by a factor of Slide10

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 3xy = -2 sin (x/3)You Try!

y = cos (-7x)y = 3 cos 2xSlide11

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:

Note: The frequency is simply the reciprocal of the period.Slide12

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

You Try!Slide13

Identify the maximum and minimum values and the zeros of the function in the interval

y = 2 sin

xSlide14

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 shiftSlide15

Combining a phase shift with a period change

Construct a sinusoid with period and

amplitude 6 that goes through (2,0) Slide16

Select the pair of functions that have identical graphs:Slide17

Select the pair of functions that have identical graphs:Slide18

Homework

Pg. 394-395

4, 12, 16, 20, 28, 33, 37, 38, 48, 54, 56, 58, 64Slide19

4.5 - Graphs of Tangent, Cotangent, Secant, and CosecantSlide20

y = tan

x

Domain:

Range:

Continuity:

Increasing/Decreasing:

Symmetry:Boundedness:

Asymptotes:End Behavior:PeriodSlide21

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.Slide22

y = cot

x

Domain:

Range:

Continuity:

Increasing/Decreasing:

Symmetry:

Boundedness:Asymptotes:End Behavior:PeriodSlide23
Slide24

Secant Function

y = sec

x

Domain:

Range:

Continuity:

Increasing/Decreasing:

Symmetry:Boundedness:Asymptotes:End Behavior:Period:Slide25

Cosecant Function

y =

csc

x

Domain:

Range:Continuity:

Increasing/Decreasing:Symmetry:Boundedness:Asymptotes:End Behavior:

Period: