Transcript PPT #1

Graphs of the
Sine and Cosine
Functions
Graphing Trigonometric
Functions
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Graph in xy-plane
Write functions as
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y
y
y
y
y
y
=
=
=
=
=
=
f(x)
f(x)
f(x)
f(x)
f(x)
f(x)
=
=
=
=
=
=
sin x
cos x
tan x
csc x
sec x
cot x
Variable x is an angle, measured in radians
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Can be any real number
Graphing the Sine Function
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Periodicity: Only need to graph on
interval [0, 2¼] (One cycle)
Plot points and graph
Properties of the Sine Function
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Domain: All real numbers
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Range: [{1, 1]
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Odd function
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Periodic, period 2¼
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x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, …
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y-intercept: 0
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Maximum value: y = 1, occurring at
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Minimum value: y = {1, occurring at
Transformations of the Graph
of the Sine Functions
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Example.
Problem: Use the graph of y = sin x to
graph
Answer:
Graphing the Cosine Function
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Periodicity: Again, only need to graph
on interval [0, 2¼] (One cycle)
Plot points and graph
Properties of the Cosine Function
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Domain: All real numbers
Range: [{1, 1]
Even function
Periodic, period 2¼
x-intercepts:
y-intercept: 1
Maximum value: y = 1, occurring at
x = …, {2¼, 0, 2¼, 4¼, 6¼, …
Minimum value: y = {1, occurring at
x = …, {¼, ¼, 3¼, 5¼, …
Transformations of the Graph
of the Cosine Functions
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Example.
Problem: Use the graph of y = cos x to
graph
Answer:
4
2
3
2
2
-2
-4
2
2
5
2
3
Sinusoidal Graphs
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Graphs of sine and cosine functions
appear to be translations of each other
Graphs are called sinusoidal
Conjecture.
Amplitude and Period of
Sinusoidal Functions
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Graphs of functions y = A sin x and y
= A cos x will always satisfy inequality
{jAj · y · jAj
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Number jAj is the amplitude
Amplitude and Period of
Sinusoidal Functions
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Graphs of functions y = A sin x and
y = A cos x will always satisfy
inequality {jAj · y · jAj
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Number jAj is the amplitude
4
4
2
2
3
2
2
2
2
5
2
3
3
2
2
-2
-2
-4
-4
2
2
5
2
3
Amplitude and Period of
Sinusoidal Functions
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Period of y = sin(!x) and
y = cos(!x) is
4
4
2
2
3
2
2
2
2
5
2
3
3
2
2
-2
-2
-4
-4
2
2
5
2
3
Amplitude and Period of
Sinusoidal Functions
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Cycle: One period of y = sin(!x) or
y = cos(!x)
4
4
2
2
3
2
2
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2
5
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-2
-2
-4
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2
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Amplitude and Period of
Sinusoidal Functions
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Cycle: One period of y = sin(!x) or
y = cos(!x)
Amplitude and Period of
Sinusoidal Functions
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Theorem. If ! > 0, the amplitude and
period of y = Asin(!x) and
y = Acos(! x) are given by
Amplitude = j Aj
Period =
.
Amplitude and Period of
Sinusoidal Functions
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Example.
Problem: Determine the amplitude and
period of y = {2cos(¼x)
Answer:
Graphing Sinusoidal Functions
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One cycle contains four important
subintervals
For y = sin x and y = cos x these are
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Gives five key points on graph
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Graphing Sinusoidal Functions
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Example.
Problem: Graph y = {3cos(2x)
Answer:
4
2
3
2
2
-2
-4
2
2
5
2
3
Finding Equations for
Sinusoidal Graphs
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Example.
Problem: Find an equation for the graph.
6
Answer:
4
2
3
5
2
2
3
2
3
2
2
-2
-4
-6
2
2
5
2
3
Key Points
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Graphing Trigonometric Functions
Graphing the Sine Function
Properties of the Sine Function
Transformations of the Graph of the
Sine Functions
Graphing the Cosine Function
Properties of the Cosine Function
Transformations of the Graph of the
Cosine Functions
Key Points (cont.)
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Sinusoidal Graphs
Amplitude and Period of Sinusoidal
Functions
Graphing Sinusoidal Functions
Finding Equations for Sinusoidal
Graphs
Graphs of the
Tangent, Cotangent,
Cosecant and Secant
Functions
Section 5.5
Graphing the Tangent
Function
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Periodicity: Only need to graph on
interval [0, ¼]
Plot points and graph
Properties of the Tangent
Function
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Domain: All real numbers, except odd
multiples of
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Range: All real numbers
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Odd function
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Periodic, period ¼
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x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, …
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y-intercept: 0
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Asymptotes occur at
Transformations of the Graph
of the Tangent Functions
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Example.
Problem: Use the graph of y = tan x to
8
graph
6
Answer:
4
2
3
2
2
-2
-4
-6
-8
2
2
5
2
3
Graphing the Cotangent Function
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Periodicity: Only need to graph on
interval [0, ¼]
Graphing the Cosecant and
Secant Functions
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Use reciprocal identities
Graph of y = csc x
Graphing the Cosecant and
Secant Functions
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Use reciprocal identities
Graph of y = sec x
Key Points
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Graphing the Tangent Function
Properties of the Tangent Function
Transformations of the Graph of the
Tangent Functions
Graphing the Cotangent Function
Graphing the Cosecant and Secant
Functions
Phase Shifts;
Sinusoidal Curve
Fitting
Graphing Sinusoidal Functions
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y = A sin(!x), ! > 0
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Amplitude jAj
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Period
y = A sin(!x { Á)
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Phase shift
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Phase shift indicates amount of shift
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To right if Á > 0
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To left if Á < 0
Graphing Sinusoidal Functions
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Graphing y = A sin(!x { Á) or
y = A cos(!x { Á):
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Determine amplitude jAj
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Determine period
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Determine starting point of one cycle:
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Determine ending point of one cycle:
Graphing Sinusoidal Functions
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Graphing y = A sin(!x { Á) or
y = A cos(!x { Á):
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Divide interval
into four
subintervals, each with length
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Use endpoints of subintervals to find the
five key points
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Fill in one cycle
Graphing Sinusoidal Functions
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Graphing y = A sin(!x { Á) or
y = A cos(!x { Á):
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Extend the graph in each direction to
make it complete
Graphing Sinusoidal Functions
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Example. For the equation
(a) Problem: Find the amplitude
Answer:
(b) Problem: Find the period
Answer:
(c) Problem: Find the phase shift
Answer:
Finding a Sinusoidal Function
from Data
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Example. An experiment in a wind tunnel generates
cyclic waves. The following data is collected for 52
seconds.
Let v represent the wind speed in feet per second and
let x represent the time in seconds.
Time (in seconds), x
Wind speed (in feet per second), v
0
21
12
42
26
67
41
40
52
20
Finding a Sinusoidal Function
from Data
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Example. (cont.)
Problem: Write a sine equation that
represents the data
Answer:
Key Points
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Graphing Sinusoidal Functions
Finding a Sinusoidal Function from
Data