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Slide Section 8.2 and

8.3 - 1 Copyright © 2009 Pearson Education, Inc.

Transformation of sine and cosine

functions

Sections 8.2 and 8.3

Revisit: Page 142; chapter 4

Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc.

Section 8.2 and 8.3

Graphs of Transformed Sine and Cosine

Functions

Graph transformations of y = sin x and y = cos x in the form y = A sin B (x – h) + k and y = A cos B (x – h) + k and determine the amplitude, the period, and the phase shift.

Graph sums of functions.

Graph functions (damped oscillations) found by multiplying trigonometric functions by other functions.

Slide Section 8.2 and

8.3 - 4 Copyright © 2009 Pearson Education, Inc.

Variations of the Basic Graphs

We are interested in the graphs of functions in

the form

y = A sin B (x – h) + k

and

y = A cos B (x – h) + k

where A, B, h, and k are all constants. These

constants have the effect of translating,

reflecting, stretching, and shrinking the basic

graphs.

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8.3 - 5 Copyright © 2009 Pearson Education, Inc.

The Constant k Let’s observe the effect of the constant k.

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The Constant k

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The Constant k

The constant D in

y = A sin B (x – h) + k

and

y = A cos B (x – h) + k

translates the graphs up k units if k > 0 or down

|k| units if k < 0.

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8.3 - 8 Copyright © 2009 Pearson Education, Inc.

The Constant A Let’s observe the effect of the constant A.

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The Constant A

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The Constant A

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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8.3 - 11 Copyright © 2009 Pearson Education, Inc.

Amplitude

The amplitude of the graphs of

is |A|.

y = A sin B (x – h) + k

and

y = A cos B (x – h) + k

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The Constant B Let’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

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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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8.3 - 17 Copyright © 2009 Pearson Education, Inc.

Period

The period of the graphs of

is

y = A sin B (x – h) + k

and

y = A cos B (x – h) + k

2

B.

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8.3 - 18 Copyright © 2009 Pearson Education, Inc.

Period: the horizontal distance between

two consecutive max/min values

The period of the graphs of

is

y = A csc B(x – h) + k

and

y = A sec B(x – h) + k

2

B.

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8.3 - 19 Copyright © 2009 Pearson Education, Inc.

Period

The period of the graphs of

is

y = A tan B(x – h) + k

and

y = A cot B(x – C) + k

B.

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The Constant h Let’s observe the effect of the constant C.

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The Constant h

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The Constant h

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The Constant h

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The Constant h

if |h| < 0, then there will be a horizontal

translation of |h| units to the right, and

if |h| > 0, then there will be a horizontal

translation of |h| units to the left.

If B = 1, then

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8.3 - 25 Copyright © 2009 Pearson Education, Inc.

Combined Transformations

B careful!

as

y = A sin (Bx – h) + k

and

y = A cos (Bx – h) + k

y Asin B x C

B

D

and

y Acos B x C

B

D

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8.3 - 26 Copyright © 2009 Pearson Education, Inc.

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

Slide Section 8.2 and

8.3 - 27 Copyright © 2009 Pearson Education, Inc.

Phase Shift

If h/B > 0, the graph is translated to the right

|h/B| units.

If h/B < 0, the graph is translated to the right

|h/B| units.

Slide Section 8.2 and

8.3 - 28 Copyright © 2009 Pearson Education, Inc.

Transformations of Sine and Cosine

Functions

To 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

Functions

1. 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

2

B.

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8.3 - 30 Copyright © 2009 Pearson Education, Inc.

Transformations of Sine and Cosine

Functions

2. 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

Slide Section 8.2 and

8.3 - 31 Copyright © 2009 Pearson Education, Inc.

Transformations of Sine and Cosine

Functions

3. Translate the graph horizontally

according to C/B.

The phase shift is C

B.

C

B 0

C

B units to the left

C

B 0

C

B units to the right

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8.3 - 32 Copyright © 2009 Pearson Education, Inc.

Transformations of Sine and Cosine

Functions

4. Translate the graph vertically according

to k.

k < 0 |k| units down

k > 0 k units up

Slide Section 8.2 and

8.3 - 33

Homework

1. Transformation of Sine Cosine functions.

2. Sec 8.2 Written exercises #1-10 all.

Copyright © 2009 Pearson Education, Inc.

Slide Section 8.2 and

8.3 - 34 Copyright © 2009 Pearson Education, Inc.

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 2

B

2

2

Phase shift C

B

2

2

4

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Example Solution 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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Example Solution continued

y sin x

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Example Solution continued

1. y sin2x

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Example Solution continued

2. y 3sin2x

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Example Solution continued 3. y 3sin 2 x

4

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Example Solution continued 4. y 3sin 2 x

4

1

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Example Graph: y = 2 sin x + sin 2x

Solution:

Graph: y = 2 sin x and y = sin 2x on the same axes.

Slide Section 8.2 and

8.3 - 42 Copyright © 2009 Pearson Education, Inc.

Example Solution continued

Graphically add some y-coordinates, or ordinates, to

obtain points on the graph that we seek.

At x = π/4, transfer h up to add it to 2 sin x, yielding P1.

At x = – π/4, transfer m down to add it to 2 sin x,

yielding P2.

At x = – 5π/4, add the negative ordinate of sin 2x to the

positive ordinate of 2 sin x, yielding P3.

This method is called addition of ordinates, because

we add the y-values (ordinates) of y = sin 2x to the y-

values (ordinates) of y = 2 sin x.

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Example Solution continued

The period of the sum 2 sin x + sin 2x is 2π, the least

common multiple of 2π and π.

Slide Section 8.2 and

8.3 - 44 Copyright © 2009 Pearson Education, Inc.

Example

Sketch a graph of f x ex 2 sin x.

Solution

f is the product of two functions g and h, where

g x ex 2 and h x sin x

To find the function values, we can multiply ordinates.

Start with 1 sin x 1

ex 2 ex 2 sin x ex 2

The graph crosses the x-axis at values of x for which sin x = 0, kπ for integer values of k.

Slide Section 8.2 and

8.3 - 45 Copyright © 2009 Pearson Education, Inc.

Example

Solution continued

f is constrained between the graphs of y = –e–x/2 and y = e–x/2. Start by graphing these functions using dashed lines.

Since f(x) = 0 when x = kπ, k an integer, we mark those points on the graph.

Use a calculator to compute other function values.

The graph is on the next slide.

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8.3 - 46 Copyright © 2009 Pearson Education, Inc.

Example

Solution continued