copyright © 2009 pearson education, inc. chapter 6: the trigonometric functions 6.1the...

Copyright © 2009 Pearson Education, Inc.

CHAPTER 6: The Trigonometric Functions

6.1 The Trigonometric Functions of Acute Angles

6.2 Applications of Right Triangles

6.3 Trigonometric Functions of Any Angle

6.4 Radians, Arc Length, and Angular Speed

6.5 Circular Functions: Graphs and Properties

6.6 Graphs of Transformed Sine and Cosine Functions

Copyright © 2009 Pearson Education, Inc.

6.6Graphs of Transformed Sine and Cosine

Functions

Graph transformations of y = sin x and y = cos x in the form y = A sin (Bx – C) + D andy = A cos (Bx – C) + D 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.

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Variations of the Basic Graphs

We are interested in the graphs of functions in the formy = 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.


The Constant DLet’s observe the effect of the constant D.


The Constant D


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.


The Constant ALet’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 (Bx – C) + D

and



The Constant BLet’s observe the effect of 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


and

y = A cos (Bx – C) + D2B

.


Period


is

y = A csc (Bx – C) + D

and

y = A sec (Bx – C) + D2B

.


Period


is

y = A tan (Bx – C) + D

and

y = A cot (Bx – C) + DB

.


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


The Constant C


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


Combined Transformations

It is helpful to rewrite

as


and


y Asin B x C

B

D

andy 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 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.


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


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

.


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


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


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

to D.

D < 0 |D| units down

D > 0 D units up


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


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



y sin x



1. y sin2x



2. y 3sin2x


ExampleSolution continued 3. y 3sin 2 x

4


ExampleSolution continued 4. y 3sin 2 x

4

1


ExampleGraph: 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 e x 2 sin x.

Solution

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

g x e x 2 and h x sin x

To find the function values, we can multiply ordinates.

Start with 1 sin x 1

e x 2 e x 2 sin x e x 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 andy = 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

copyright © 2009 pearson education, inc. chapter 6: the trigonometric functions 6.1the...

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