Post on 30-Nov-2014
description
73 Periodic Graphs and Amplitude
copy 2008 Roy L Gover(wwwmrgovercom)
Learning GoalsbullUnderstand period and amplitude of a Trigonometric functionbullIntroduce real world applications of periodic graphs
Important IdeaPeriod and amplitude are important in describing the nature of many real world phenomenon such as light waves sound waves and radio waves
Larger PeriodSmaller Frequency
Important Idea
Radio Visible X-Ray
Smaller PeriodGreater Frequency
Important Idea
Radio Visible X-Ray
Important Idea
Radio Waves
Important Idea
Medicine
Important Idea
Ocean Tides
Definition
A complete waveform of a periodic function with no repeats is a cycle
Important IdeaA cycle may be measured in different ways but for a given function the period will be the same
Period
Definition
The Period is the time required for the function to repeat
t
( )f t
Go to Sketchpad
Definition
Frequency (in cycles per second) is the reciprocal of the period
Start Movie
Try ThisbullWhat are your observations about the movement of the bridge
bullWhat do you think is the connection between this bridge and todayrsquos lesson
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Important IdeaPeriod and amplitude are important in describing the nature of many real world phenomenon such as light waves sound waves and radio waves
Larger PeriodSmaller Frequency
Important Idea
Radio Visible X-Ray
Smaller PeriodGreater Frequency
Important Idea
Radio Visible X-Ray
Important Idea
Radio Waves
Important Idea
Medicine
Important Idea
Ocean Tides
Definition
A complete waveform of a periodic function with no repeats is a cycle
Important IdeaA cycle may be measured in different ways but for a given function the period will be the same
Period
Definition
The Period is the time required for the function to repeat
t
( )f t
Go to Sketchpad
Definition
Frequency (in cycles per second) is the reciprocal of the period
Start Movie
Try ThisbullWhat are your observations about the movement of the bridge
bullWhat do you think is the connection between this bridge and todayrsquos lesson
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Larger PeriodSmaller Frequency
Important Idea
Radio Visible X-Ray
Smaller PeriodGreater Frequency
Important Idea
Radio Visible X-Ray
Important Idea
Radio Waves
Important Idea
Medicine
Important Idea
Ocean Tides
Definition
A complete waveform of a periodic function with no repeats is a cycle
Important IdeaA cycle may be measured in different ways but for a given function the period will be the same
Period
Definition
The Period is the time required for the function to repeat
t
( )f t
Go to Sketchpad
Definition
Frequency (in cycles per second) is the reciprocal of the period
Start Movie
Try ThisbullWhat are your observations about the movement of the bridge
bullWhat do you think is the connection between this bridge and todayrsquos lesson
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Smaller PeriodGreater Frequency
Important Idea
Radio Visible X-Ray
Important Idea
Radio Waves
Important Idea
Medicine
Important Idea
Ocean Tides
Definition
A complete waveform of a periodic function with no repeats is a cycle
Important IdeaA cycle may be measured in different ways but for a given function the period will be the same
Period
Definition
The Period is the time required for the function to repeat
t
( )f t
Go to Sketchpad
Definition
Frequency (in cycles per second) is the reciprocal of the period
Start Movie
Try ThisbullWhat are your observations about the movement of the bridge
bullWhat do you think is the connection between this bridge and todayrsquos lesson
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Important Idea
Radio Waves
Important Idea
Medicine
Important Idea
Ocean Tides
Definition
A complete waveform of a periodic function with no repeats is a cycle
Important IdeaA cycle may be measured in different ways but for a given function the period will be the same
Period
Definition
The Period is the time required for the function to repeat
t
( )f t
Go to Sketchpad
Definition
Frequency (in cycles per second) is the reciprocal of the period
Start Movie
Try ThisbullWhat are your observations about the movement of the bridge
bullWhat do you think is the connection between this bridge and todayrsquos lesson
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Important Idea
Medicine
Important Idea
Ocean Tides
Definition
A complete waveform of a periodic function with no repeats is a cycle
Important IdeaA cycle may be measured in different ways but for a given function the period will be the same
Period
Definition
The Period is the time required for the function to repeat
t
( )f t
Go to Sketchpad
Definition
Frequency (in cycles per second) is the reciprocal of the period
Start Movie
Try ThisbullWhat are your observations about the movement of the bridge
bullWhat do you think is the connection between this bridge and todayrsquos lesson
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Important Idea
Ocean Tides
Definition
A complete waveform of a periodic function with no repeats is a cycle
Important IdeaA cycle may be measured in different ways but for a given function the period will be the same
Period
Definition
The Period is the time required for the function to repeat
t
( )f t
Go to Sketchpad
Definition
Frequency (in cycles per second) is the reciprocal of the period
Start Movie
Try ThisbullWhat are your observations about the movement of the bridge
bullWhat do you think is the connection between this bridge and todayrsquos lesson
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Definition
A complete waveform of a periodic function with no repeats is a cycle
Important IdeaA cycle may be measured in different ways but for a given function the period will be the same
Period
Definition
The Period is the time required for the function to repeat
t
( )f t
Go to Sketchpad
Definition
Frequency (in cycles per second) is the reciprocal of the period
Start Movie
Try ThisbullWhat are your observations about the movement of the bridge
bullWhat do you think is the connection between this bridge and todayrsquos lesson
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Important IdeaA cycle may be measured in different ways but for a given function the period will be the same
Period
Definition
The Period is the time required for the function to repeat
t
( )f t
Go to Sketchpad
Definition
Frequency (in cycles per second) is the reciprocal of the period
Start Movie
Try ThisbullWhat are your observations about the movement of the bridge
bullWhat do you think is the connection between this bridge and todayrsquos lesson
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Definition
The Period is the time required for the function to repeat
t
( )f t
Go to Sketchpad
Definition
Frequency (in cycles per second) is the reciprocal of the period
Start Movie
Try ThisbullWhat are your observations about the movement of the bridge
bullWhat do you think is the connection between this bridge and todayrsquos lesson
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Definition
Frequency (in cycles per second) is the reciprocal of the period
Start Movie
Try ThisbullWhat are your observations about the movement of the bridge
bullWhat do you think is the connection between this bridge and todayrsquos lesson
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Start Movie
Try ThisbullWhat are your observations about the movement of the bridge
bullWhat do you think is the connection between this bridge and todayrsquos lesson
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try ThisbullWhat are your observations about the movement of the bridge
bullWhat do you think is the connection between this bridge and todayrsquos lesson
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Important Idea
The x-axis is now the t-axis (time) The y-axis is now the f (t) axis
t
( )f t
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Important IdeaFor the trigonometric functions
( ) sinf t bt( ) cosg t bt
or
The constant b changes the period
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
DefinitionThe trigonometric functions
( ) sinf t bt( ) cosg t bt
have periods (repeat) every 2
b
radians
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
ExampleWhat is the period of
( ) sinf t t
02
Period=2
21
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
ExampleWhat is the period of
( ) sin(2 )g t t0 2
2
2
Period=
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
ExampleWhat is the period of
( ) sin(5 )h t t
Period=2
45
0 4 8
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Important Idea
( ) sin( )f t x
( ) sin(2 )g t t
( ) sin(5 )h t t0 2 4
As b gets small period gets large
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try ThisWhat is the period (how often does the function repeat) ( ) cosp t t
22
1
0 2 4
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try ThisWhat is the period (how often does the function repeat) ( ) cos12p t t
2167
12
0 167
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try This
Which is the graph of ( ) sin(4 )f t t 0
red2
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try ThisWhat is the equation of the function graphed in blue( ) sin(2 )f t t
02
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Important IdeaThe trigonometric function( ) tanf t bt
has a period (repeats) every b
radians
t
( )f t
b
b
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
ExampleDetermine how often the function repeats (period) ( ) tan(2 )k t t
( ) tan3
tj t
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try ThisDetermine how often the function repeats (period)
( ) tan6
tp t
1
6b 6 period=
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try ThisUsing your calculator graph
1 tany t
2 tan6
ty
Compare and contrast
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
DefinitionAmplitude is where is the distance from the middle to the top and
aa
ais the distance from the middle to the bottom
aa
Geom Sketchpad
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try ThisCompare and Contrast( ) 5sinf t t( ) 2sing t t
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try ThisWhat is the amplitude of ( ) 6 tanp t tHint this is a trick question
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Important Idea( ) sinf t a bt ( ) cosg t a bt
The constant a controls the amplitude (vertical stretch) The constant b controls the horizontal stretch Amplitude is
Period is
a2
b
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try ThisFind the amplitude and period for ( ) sinf t t
Amplitude=1 Period =2
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try ThisFind the amplitude and period for ( ) 3cos2f t t
Amplitude=3 Period =
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try ThisFind the amplitude and period for ( ) 3tan(5 )f t t
Amplitude None Period=2
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Example
For what value(s) of t between is
sin 2 1t 0 1t
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Example
For what value(s) of t between [0 ] is
cos2 0t 2
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Example
For what value(s) of t between [0 ] is
cos3 0t 2
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try This
For what value(s) of t between [0 ] is
sin3 0t
03
t t
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Try This
For what value(s) of t between [01] is
sin3 0t 1 2
0 13 3
t
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84
Lesson ClosePeriod and amplitude will be important when we study modeling and simple harmonic motion in section 84