Sinusoids and Sinusoids and TransformationsTransformations

Sec. 4.4bSec. 4.4b

Definition: Sinusoid

sin[ ( )]f x a b x h k

A function is a sinusoid if it can be written in the form

where a, b, c, and d are constants and neither a nor b is 0.

In general, any transformation of a sine function (or the graphof such a function – such as cosine) is a sinusoid.

sin( )f x a bx c d

This is the format that we are used to seeing, thus it is OK to continue using this format…I use this format.

TransformationsThere is a special vocabulary for describing our traditionalgraphical transformations when applied to sinusoids…

Horizontal stretches and shrinks affect the periodand the frequency.

Vertical stretches and shrinks affect the amplitude.

Horizontal translations bring about phase shifts.

Definition: Amplitude of a Sinusoid

sin[ ( )]f x a b x h k The amplitude of the sinusoid

is a

Similarly, the amplitude of

cos[ ( )]f x a b x h k is a

Graphically, the amplitude is half the height of the wave.

TransformationsFind the amplitude of each function and use the language oftransformations to describe how the graphs are related.

(a)1 cosy x (b) 2

1cos2

y x (c) 3 3cosy x

Amplitudes: (a) 1, (b) 1/2, (c) |–3| = 3

The graph of y is a vertical shrink of the graph of y by 1/2.2 1

The graph of y is a vertical stretch of the graph of y by 3,and a reflection across the x-axis, performed in either order.

3 1

Confirm these answers graphically!!!Confirm these answers graphically!!!

Definition: Period of a Sinusoid

sin[ ( )]f x a b x h k The period of the sinusoid

is 2 b

Similarly, the period of

cos[ ( )]f x a b x h k is2 b

Graphically, the period is the length of one full cycle ofthe wave.

TransformationsFind the period of each function and use the language oftransformations to describe how the graphs are related.

(a)1 siny x

(b) 2 2sin3

xy

(c) 3 3sin 2y x

Periods

2

2 1 3 6

2 2

TransformationsFind the period of each function and use the language oftransformations to describe how the graphs are related.

(a)1 siny x

(b) 2 2sin3

xy

(c) 3 3sin 2y x

The graph of y is a horizontalstretch of the graph of y by 3, avertical stretch by 2, and areflection across the x-axis,performed in any order.

2

1

TransformationsFind the period of each function and use the language oftransformations to describe how the graphs are related.

(a)1 siny x

(b) 2 2sin3

xy

(c) 3 3sin 2y x

The graph of y is a horizontalshrink of the graph of y by 1/2, avertical stretch by 3, and areflection across the y-axis,performed in any order.

3

1

Confirm these answers Confirm these answers graphically!!!graphically!!!

Definition: Frequency of a Sinusoid

sin[ ( )]f x a b x h k The frequency of the sinusoid

is 2b

Similarly, the frequency of

cos[ ( )]f x a b x h k is 2b Graphically, the frequency is the number of complete cyclesthe wave completes in a unit interval.

Reciprocal of the period!!!

TransformationsFind the frequency of the given function, and interpret itsmeaning graphically.

24sin

3

xf x

Sketch the graph of the function in by 2 ,2 4,4

Frequency:2 3 1

2 3

Period: 3The graph completes 1 full cycleper interval of length .3

Interpretation:

TransformationsHow does the graph of differ from thegraph of ?

y f x c y f x

A translation to the left by c units when c > 0

New Terminology: When applied to sinusoids, wesay that the wave undergoes a phase shift of –c.

TransformationsWrite the cosine function as a phase shift of the sine function.

cos sin 2x x Write the sine function as a phase shift of the cosine function.

sin cos 2x x

Confirm these answers graphically!!!Confirm these answers graphically!!!

Finally, a couple of whiteboard problems

Find the amplitude of the function and use the language oftransformations to describe how the graph of the function isrelated to the graph of the sine function.

2siny x1. Amplitude 2; Vertical stretch by 2Amplitude 2; Vertical stretch by 2

4siny x3. Amplitude 4; Vertical stretch by 4,Amplitude 4; Vertical stretch by 4,Reflect across Reflect across xx-axis-axis

More whiteboard…Find the period of the function and use the language oftransformations to describe how the graph of the function isrelated to the graph of the cosine function.

cos3y x7. Period ; Horizontal shrinkPeriod ; Horizontal shrinkby 1/3by 1/3

cos 7y x 9. Period ; HorizontalPeriod ; Horizontalshrink by 1/7, Reflect acrossshrink by 1/7, Reflect acrossyy-axis-axis

2 3

2 7

Reminder: Graphs of Sinusoids

The graphs of these functions have the following characteristics:

siny a b x h k cosy a b x h k

0, 0a b

Amplitude = a Period =2

b

Frequency =

2

b

A phase shift of h A vertical translation of k

Guided PracticeGraph one period of the given function by hand.

2.5siny x Amplitude = 2.5 Period = 2

by , 3,3

Guided PracticeGraph one period of the given function by hand.

4cosy x Amplitude = 4 Period = 2

by , 4,4

WhiteboardGraph three periods of the given function by hand.

3cos 2y x Amplitude = 3 Period = 4

by 6 ,6 3,3

WhiteboardGraph three periods of the given function by hand.

20sin 4y x Amplitude = 20 Period =2

by 3 4,3 4 30,30

WhiteboardGraph three periods of the given function by hand.

8cos5y x Amplitude = 8 Period =2

5

by 3 5,3 5 9,9

Guided PracticeIdentify the maximum and minimum values and the zeros of thegiven function in the interval no calculator! 2 ,2

Maximum:

3cos2

xy

3 At 0

Minimum: 3 At 2Zeros:

WhiteboardIdentify the maximum and minimum values and the zeros of thegiven function in the interval no calculator! 2 ,2

Maximum:

0.5siny x1

2At

3,

2 2

Minimum:1

2 At

2

and

3

2

Zeros: 0, , 2

WhiteboardState the amplitude and period of the given sinusoid, and(relative to the basic function) the phase shift and verticaltranslation.

3.5sin 2 12

y x

Amplitude:3.5 Period: Phase Shift:4

Vertical Translation: 1 unit down

3.5sin 2 14

x

WhiteboardState the amplitude and period of the given sinusoid, and(relative to the basic function) the phase shift and verticaltranslation. 2 3

cos 73 4

xy

Amplitude:2

3Period: 8 Phase Shift: 3

Vertical Translation: 7 units up

2 1cos 3 73 4

x

Practice ProblemsPractice ProblemsConstruct a sinusoid with period and amplitude 6 that goesthrough (2,0).

5

10b

siny a b x h k

2

5b

First, solve for b:

Either will work!!!

6a Find the amplitude:

6a Let’s just take the positive value again.

To pass through (2,0), we need a phaseshift of 2 h = –2

6sin 10 2 6sin 10 20y x x

Practice ProblemsPractice ProblemsConstruct a sinusoid y = f(x) that rises from a minimum value ofy = 5 at x = 0 to a maximum value of y = 25 at x = 32.

First, sketch a graph of this sinusoid…

25 510

2a

Amplitude is half the height:

The period is 64: 264

b

32b

We need a function whose minimum is at x = 0. We could shiftthe sine function horizontally, but it’s easier to simply reflect thecosine function……by letting a = –10

Practice ProblemsPractice ProblemsConstruct a sinusoid y = f(x) that rises from a minimum value ofy = 5 at x = 0 to a maximum value of y = 25 at x = 32.

10cos32

y x

But since cosine is aneven function:

This function ranges from –10 to 10, but we need a functionthat ranges from 5 to 25………vertical translation by 15:

10cos32x

10cos 1532

y x

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graphically???graphically???

Amplitude: 2, Period: , Point (0,0)

Practice ProblemsPractice ProblemsConstruct a sinusoid with the given information.

3

2sin 2 3y xOne possibility:

Amplitude: 3.2, Period: , Point (5,0)

Practice ProblemsPractice ProblemsConstruct a sinusoid with the given information.

7

3.2sin 14 5y x One possibility: