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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.
Slide Section 8.2 and
8.3 - 5 Copyright © 2009 Pearson Education, Inc.
The Constant k Let’s observe the effect of the constant k.
Slide Section 8.2 and
8.3 - 6 Copyright © 2009 Pearson Education, Inc.
The Constant k
Slide Section 8.2 and
8.3 - 7 Copyright © 2009 Pearson Education, Inc.
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.
Slide Section 8.2 and
8.3 - 8 Copyright © 2009 Pearson Education, Inc.
The Constant A Let’s observe the effect of the constant A.
Slide Section 8.2 and
8.3 - 9 Copyright © 2009 Pearson Education, Inc.
The Constant A
Slide Section 8.2 and
8.3 - 10 Copyright © 2009 Pearson Education, Inc.
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.
Slide Section 8.2 and
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
Slide Section 8.2 and
8.3 - 12 Copyright © 2009 Pearson Education, Inc.
The Constant B Let’s observe the effect of the constant B.
Slide Section 8.2 and
8.3 - 13 Copyright © 2009 Pearson Education, Inc.
The Constant B
Slide Section 8.2 and
8.3 - 14 Copyright © 2009 Pearson Education, Inc.
The Constant B
Slide Section 8.2 and
8.3 - 15 Copyright © 2009 Pearson Education, Inc.
The Constant B
Slide Section 8.2 and
8.3 - 16 Copyright © 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.
Slide Section 8.2 and
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.
Slide Section 8.2 and
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.
Slide Section 8.2 and
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.
Slide Section 8.2 and
8.3 - 20 Copyright © 2009 Pearson Education, Inc.
The Constant h Let’s observe the effect of the constant C.
Slide Section 8.2 and
8.3 - 21 Copyright © 2009 Pearson Education, Inc.
The Constant h
Slide Section 8.2 and
8.3 - 22 Copyright © 2009 Pearson Education, Inc.
The Constant h
Slide Section 8.2 and
8.3 - 23 Copyright © 2009 Pearson Education, Inc.
The Constant h
Slide Section 8.2 and
8.3 - 24 Copyright © 2009 Pearson Education, Inc.
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
Slide Section 8.2 and
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
Slide Section 8.2 and
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
Slide Section 8.2 and
8.3 - 29 Copyright © 2009 Pearson Education, Inc.
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.
Slide Section 8.2 and
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
Slide Section 8.2 and
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
Slide Section 8.2 and
8.3 - 35 Copyright © 2009 Pearson Education, Inc.
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
Slide Section 8.2 and
8.3 - 36 Copyright © 2009 Pearson Education, Inc.
Example Solution continued
y sin x
Slide Section 8.2 and
8.3 - 37 Copyright © 2009 Pearson Education, Inc.
Example Solution continued
1. y sin2x
Slide Section 8.2 and
8.3 - 38 Copyright © 2009 Pearson Education, Inc.
Example Solution continued
2. y 3sin2x
Slide Section 8.2 and
8.3 - 39 Copyright © 2009 Pearson Education, Inc.
Example Solution continued 3. y 3sin 2 x
4
Slide Section 8.2 and
8.3 - 40 Copyright © 2009 Pearson Education, Inc.
Example Solution continued 4. y 3sin 2 x
4
1
Slide Section 8.2 and
8.3 - 41 Copyright © 2009 Pearson Education, Inc.
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.
Slide Section 8.2 and
8.3 - 43 Copyright © 2009 Pearson Education, Inc.
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.
Slide Section 8.2 and
8.3 - 46 Copyright © 2009 Pearson Education, Inc.
Example
Solution continued