5.5 a&b Graphs of Sine and Cosine HW p. 589 1-23 odd, 31-65 odd

5.5 a&b Graphs of Sine and Cosine

HW p. 589 1-23 odd, 31-65 odd

Graphical Analysis: graph the function in the given window

2 ,2 by 3,3

Domain:

sinf x x

, Range: 1,1Continuous

Alternately increasing and decreasingin periodic waves

Symmetry: Origin (odd function)

Bounded Absolute Max. of 1 Absolute Min. of –1

No Horizontal Asymptotes No Vertical Asymptotes

lim sinx

x

End Behavior: limsinx

x

and do not exist

The “Do Now” – first, graph the function in the given window

2 ,2 by 3,3

Other notes:

sinf x x

• This function is periodic, with period 2• By definition, sin(t) is the y-coordinate of the point P on the unit circle to which the real number t gets wrapped

So now let’s “explore” where this wavy graph comes from…

Now, a Complete Analysis of the Cosine Function

2 ,2 by 3,3

Domain:

cosf x x

, Range: 1,1Continuous

Alternately increasing and decreasingin periodic waves

Symmetry: y-axis (even function)

Bounded Absolute Max. of 1 Absolute Min. of –1

No Horizontal Asymptotes No Vertical Asymptotes

lim cosx

x

End Behavior: limcosx

x

and do not exist

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!!!

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


(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


(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


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


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

A phase shift of h A vertical translation of k

Guided PracticeGraph one period of the given function by hand.

2.5siny x

Amplitude = 2.5Period = 2

Guided PracticeGraph one period of the given function by hand.

4cosy x

Amplitude = 4Period = 2

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:

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 2

4siny x3. Amplitude 4; Vertical stretch by 4,Reflect across x-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 shrink by 1/3

cos 7y x 9. Period Horizontal shrink by 1/7, Reflect across y-axis

2 3

2 7

WhiteboardGraph three periods of the given function by hand.

3cos 2y x Amplitude = 3 Period = 4


20sin 4y x Amplitude = 20 Period =2


8cos5y x Amplitude = 8 Period =2

5

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

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

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5.5 a&b Graphs of Sine and Cosine HW p. 589 1-23 odd, 31-65 odd