Basic Graphs of Sine and Cosine Functions 4'1

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Basic Graphs of Sine and Cosine Functions4.1

• JMerrill, 2009
• (contributions by DDillon)

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Sine Function
0
1
0
-1
0
Notice the sine function has origin symmetry.
(If you rotate it 180 about the origin, the
graph looks the same.) This means that the sine
function is odd. sin (-x) - sin x
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Period Sine Function
0
1
0
-1
0
This one piece of the sine function repeats over
and over, causing the sine function to be
periodic. The length of this piece is called
the period of the function.
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Cosine Function
1
0
1
0
-1
Notice the cosine function has y-axis symmetry.
(If you reflect it across the y-axis, the graph
looks the same.) This means that the cosine
function is even. cos (-x) cos x
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Period Cosine Function
1
0
1
0
-1
This one piece of the cosine function repeats
over and over, causing the cosine function to be
periodic. The length of this piece is called the
period of the function.
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Period
The period of a normal sine or cosine function is
2p. To change the period of a sine or cosine
function, you would need to horizontally stretch
or shrink the function. The period is found by
period
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PeriodExamples of f(x) sin Bx

• The period of the sin(x) (parent) is 2p
• The period of sin2x is p. p
• If B gt 1, the graph shrinks.
• This graph is happening twice as often as the
  original wave.

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PeriodExamples of f(x) sin Bx

• The period of the sinx (parent) is 2p
• The period of sin ½ x is 4p. p
• If b lt 1, the graph stretches.
• This graph is happening half as often as the
  original wave.

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What is the period? Examples
Horiz. stretch by ½
Horiz. shrink by 3
Horiz. shrink by 2p/3
Horiz. shrink by p/2
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Amplitude Sine Function
0
1
0
-1
0
The maximum height of the sine function is 1. It
goes one unit above and one unit below the
x-axis, which is the center of its graph. This
maximum height is called the amplitude.
1
1
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Amplitude Cosine Function
1
0
1
0
-1
The maximum height of the cosine function is 1.
It goes one unit above and one unit below the
x-axis, which is the center of its graph. This
maximum height is called the amplitude.
1
1
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Amplitude
The amplitude of the normal sine or cosine
function is 1. To change the amplitude of a sine
or cosine function, you would need to vertically
stretch or shrink the function. amplitude
A (Choose the line that is dead-center of the
graph. The amplitude has the same height above
the center line (axis of the wave) as the height
below the center line.
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What is the amplitude? Examples
Vert. stretch by 3
Vert. shrink by ½
Vert. shrink by p/4
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Examples y A sin Bx y A cos Bx

• Give the amplitude and period of each funtion
• Y 4 cos 2x
• A 4,
• y -4 sin 1/3 x
• A 4,

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Can You Write the Equation?

• Sine or cosine?
• Amplitude?
• Period?
• b?
• Equation?

2
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Equation?

• Sine or Cosine?
• Amplitude?
• Period?
• b?
• Equation

2
8
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Harmonic Motion

• 3 Types
• Simple unvarying period motion
• Damped motion decreases with time
• Resonance motion increases with time

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Weight on Spring
video
A weight is at rest hanging from a spring. It is
then pulled down 6 cm and released. The weight
oscillates up and down, completing one cycle
every 3 seconds.
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Sketch
Distance above/below resting point, in cm
6
Time, in seconds
3
-6
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Equation
Amplitude 6 A 6 3 2p/B B 2p/3
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Positions
Determine the position of the weight at 1.5
seconds. Let x 1.5 plug into equation for
function. y 0 cm (back at original position)
Use the graph to find the time when y 3.5 for
the first time. Graph y1 equation you wrote
graph y2 3.5. Find intersection. x 1.797
seconds 3.5 is the 3.5 cm distance above the
original position of the weight.