slide 6.6 - 1 graphs transformation of sine and cosine consider the form y = a sin (bx – c) + d...

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Graphs Transformation of Sine and Cosine

Consider the form

y = A sin (Bx – C) + D

and

y = A cos (Bx – C) + D

where A, B, C, and D are all constants. These constants have the effect of translating, reflecting, stretching, and shrinking the basic graphs.

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Let’s observe the effect of the constant D.

Vertical Shift

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Vertical Shift

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The Constant D

The constant D iny = A sin (Bx – C) + D

and


translates the graphs up D units if D > 0 or down |D| units if D < 0.

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The AmplitudeThe amplitude of the graphs of

Let’s observe the effect of the constant A.

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The Amplitude

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The Constant |A| is the amplitude of the graph

If |A| > 1, then there will be a vertical stretching.

If |A| < 1, then there will be a vertical shrinking.

If A < 0, the graph is also reflected across the x-axis.

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The Constant BLet’s observe the effect of the constant B.

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The Constant B

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The Constant B

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The Constant B

6 - 12Copyright © 2009 Pearson Education, Inc.

The Constant B

If |B| < 1, then there will be a horizontal stretching.

If |B| > 1, then there will be a horizontal shrinking.

If B < 0, the graph is also reflected across the y-axis.

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Period

The period of the graphs of

is


and

y = A cos (Bx – C) + D2B

.

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Period


is

y = A csc (Bx – C) + D

and

y = A sec (Bx – C) + D2B

.

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Period


is

y = A tan (Bx – C) + D

and

y = A cot (Bx – C) + DB

.

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The Constant CLet’s observe the effect of the constant C.

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The Constant C

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The Constant C

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The Constant C

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The Constant C

if |C| < 0, then there will be a horizontal translation of |C| units to the right, and

if |C| > 0, then there will be a horizontal translation of |C| units to the left.

If B = 1, then

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Combined Transformations

It is helpful to rewrite

as


and


y Asin B x C

B

D

andy Acos B x

C

B

D

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Phase Shift

The phase shift of the graphs

is the quantity

and

C

B.

y Asin Bx C D Asin B x C

B

D

y Acos Bx C D Acos B x C

B

D

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Phase Shift

If C/B > 0, the graph is translated to the right |C/B| units.

If C/B < 0, the graph is translated to the right |C/B| units.

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Transformations of Sine and Cosine FunctionsTo graph

follow the steps listed below in the order in which they are listed.

and

y Asin Bx C D Asin B x C

B

D

y Acos Bx C D Acos B x C

B

D

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Transformations of Sine and Cosine Functions1. Stretch or shrink the graph horizontally

according to B.

The period is

|B| < 1 Stretch horizontally

|B| > 1 Shrink horizontally

B < 0 Reflect across the y-axis

2B

.

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Transformations of Sine and Cosine Functions2. Stretch or shrink the graph vertically

according to A.

The amplitude is A.

|A| < 1 Shrink vertically

|A| > 1 Stretch vertically

A < 0 Reflect across the x-axis

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Transformations of Sine and Cosine Functions3. Translate the graph horizontally

according to C/B.

The phase shift isC

B.

C

B 0

C

B units to the left

C

B 0

C

B units to the right

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Transformations of Sine and Cosine Functions4. Translate the graph vertically according

to D.

D < 0 |D| units down

D > 0 D units up

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Example

Sketch the graph of

Solution:

y 3sin 2x / 2 1.

Find the amplitude, the period, and the phase shift.

y 3sin 2x 2

1 3sin 2 x

4

1

Amplitude A 3 3

Period 2B

22

Phase shift C

B

2

2

4

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ExampleSolution continued

1. y sin2x

Then we sketch graphs of each of the following equations in sequence.

4. y 3sin 2 x 4

1

To create the final graph, we begin with the basic sine curve, y = sin x.

2. y 3sin2x

3. y 3sin 2 x 4

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y sin x

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1. y sin2x

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2. y 3sin2x

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ExampleSolution continued 3. y 3sin 2 x

4

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ExampleSolution continued 4. y 3sin 2 x

4

1

slide 6.6 - 1 graphs transformation of sine and cosine consider the form y = a sin (bx – c) + d...

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