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2. Transformation of sine and cosine functions
Sections 8.2 and 8.3
Revisit: Page 142; chapter 4
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3. 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.
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4. 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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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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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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Amplitude
The amplitude of the graphs of
y = A sin B (x – h) + k
and
y = A cos B (x – h) + k
is |A|.
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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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Period
The period of the graphs of
y = A sin B (x – h) + k
and
y = A cos B (x – h) + k is
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18. Period: the horizontal distance between two consecutive max/min values
The period of the graphs of
y = A csc B(x – h) + k
and
y = A sec B(x – h) + k is
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Period
The period of the graphs of
y = A tan B(x – h) + k
and
y = A cot B(x – C) + k is
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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 B = 1, then
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.
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25. Combined Transformations
B careful!
y = A sin (Bx – h) + k
and
y = A cos (Bx – h) + k as
and
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Phase Shift
The phase shift of the graphs
and
is the quantity
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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.
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28. Transformations of Sine and Cosine Functions
To graph
and
follow the steps listed below in the order in which they are listed.
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29. Transformations of Sine and Cosine Functions
1. Stretch or shrink the graph horizontally according to B.
|B| < Stretch horizontally
|B| > Shrink horizontally
B < Reflect across the y-axis
The period is
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30. Transformations of Sine and Cosine Functions
2. Stretch or shrink the graph vertically according to A.
|A| < Shrink vertically
|A| > Stretch vertically
A < Reflect across the x-axis
The amplitude is A.
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31. Transformations of Sine and Cosine Functions
3. Translate the graph horizontally according to C/B.
The phase shift is
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32. Transformations of Sine and Cosine Functions
4. Translate the graph vertically according to k.
k < |k| units down
k > k units up
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Homework
1. Transformation of Sine Cosine functions.
Sec 8.2 Written exercises #1-10 all.
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Example
Sketch the graph of
Find the amplitude, the period, and the phase shift.
Solution:
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Example
Solution continued
To create the final graph, we begin with the basic sine curve, y = sin x.
Then we sketch graphs of each of the following equations in sequence.
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Example
Solution continued
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Example
Solution continued
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Example
Solution continued
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Example
Solution continued
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Example
Solution continued
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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.
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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.
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Example
Sketch a graph of
Solution
f is the product of two functions g and h, where
To find the function values, we can multiply ordinates.
Start with
The graph crosses the x-axis at values of x for which sin x = 0, kπ for integer values of k.
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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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Example
Solution continued
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