Slide Section 8.2 and 8.3 - 1 - WordPress.com · Copyright © 2009 Pearson Education, Inc. Section 8.2 and 8.3 Graphs of Transformed Sine and Cosine Functions Graph transformations

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.


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.



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.



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



The Constant k



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.



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



The Constant A



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.



Amplitude

The amplitude of the graphs of

is |A|.

y = A sin B (x – h) + k

and

y = A cos B (x – h) + k



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



The Constant B



The Constant B



The Constant B



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.



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.



Period: the horizontal distance between

two consecutive max/min values


is

y = A csc B(x – h) + k

and

y = A sec B(x – h) + k

2

B.



Period


is

y = A tan B(x – h) + k

and

y = A cot B(x – C) + k

B.



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



The Constant h



The Constant h



The Constant h



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



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



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



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.



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




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.




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




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




Functions

4. Translate the graph vertically according

to k.

k < 0 |k| units down

k > 0 k units up


8.3 - 33

Homework

1. Transformation of Sine Cosine functions.

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




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



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




y sin x




1. y sin2x




2. y 3sin2x



Example Solution continued 3. y 3sin 2 x

4



Example Solution continued 4. y 3sin 2 x

4

1



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

Solution:

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




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.




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

common multiple of 2π and π.



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.



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.



Example

Solution continued

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