Quiz 4-3 1.Find a positive and negative co-terminal angle with: co-terminal angle with: 2.Find a positive and negative co-terminal angle with: co-terminal.

Quiz 4-3Quiz 4-3

6

1.1. Find a positive Find a positive andand negative negative co-terminal angle with: co-terminal angle with:

2.2. Find a positive Find a positive andand negative negative co-terminal angle with: co-terminal angle with:

100º100º

3. A ray passes through the point (4,5). Find 3. A ray passes through the point (4,5). Find ?csc

4. Find 4. Find tan 300tan 300ºº without a calculator. without a calculator.

Mini-TESTMini-TEST1. 1. On a unit circle, label all of the “nice angles” that come On a unit circle, label all of the “nice angles” that come

from the 45-45-90 and 30-60-90 right triangles in both from the 45-45-90 and 30-60-90 right triangles in both degreesdegrees and and radiansradians..

2. 2. Label the x and y coordinates for each of the points on Label the x and y coordinates for each of the points on the circle where the terminal side of the above angles the circle where the terminal side of the above angles intersect the circle.intersect the circle.

Without using a calculator find the following:Without using a calculator find the following:

3. 3. sec 45sec 45

4. 4. Cos 225Cos 225

5. 5. Csc 3Csc 3ππ/2/2

6. 6. Tan Tan ππ

4.44.4 Graphs of Sine and Cosine: Graphs of Sine and Cosine: SinusoidsSinusoids

What you’ll learn:What you’ll learn: The Basic Waves RevisitedThe Basic Waves Revisited Sinusoids and TransformationsSinusoids and Transformations Modeling Periodic Behavior with SinusoidsModeling Periodic Behavior with Sinusoids

… … and whyand why

Sine and cosine gain added significance when Sine and cosine gain added significance when used to model waves and periodic behavior.used to model waves and periodic behavior.

Graph of the Sine Function Graph of the Sine Function f (x) = sin xf (x) = sin x

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7

radians

radians

Think of a dot traveling around the circle to the right.Think of a dot traveling around the circle to the right.

At the exact same time, a corresponding At the exact same time, a corresponding dot is traveling along the x-axis to the left.dot is traveling along the x-axis to the left.

Graph of the Sine Function Graph of the Sine Function f (x) = sin xf (x) = sin x

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7

radians

radians

Think of ‘x’ as either (1) the central angle or Think of ‘x’ as either (1) the central angle or (2) the distance around the circle starting at the + x-axis.(2) the distance around the circle starting at the + x-axis. if radius =1 then angle measure in radians = arc length.if radius =1 then angle measure in radians = arc length.

Graph of the Sine Function: f (x) = sin Graph of the Sine Function: f (x) = sin xx

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7

radians

radians

Think of Think of sin xsin x as the distance as the distance aboveabove (or below) the x-axis (or below) the x-axis

determined by ‘t’ (the distance around the circle). determined by ‘t’ (the distance around the circle).

SinusoidSinusoid

Your turn:Your turn:1. 1. Domain = ?Domain = ?2. 2. Range = ?Range = ?3. 3. Continuous? Continuous?4. 4. Symmetry? Symmetry?5. 5. bounded? bounded?6. 6. Vertical/horizontal asymptotes? Vertical/horizontal asymptotes?7. 7. End behavior? End behavior?

SinusoidSinusoid A function is a if it can be written in the form

( ) sin( ) where , , , and are constants

and neither nor is 0.

f x a bx c d a b c d

a b

sinusoid

SinusoidSinusoid dcbxaxf sin)(

Vertical stretch/shrinkVertical stretch/shrink

If a < 0: reflection across x-axisIf a < 0: reflection across x-axis

Horizontal stretch/shrink by factor of: Horizontal stretch/shrink by factor of: b

1

If b < 0: reflection across y-axisIf b < 0: reflection across y-axis

Horizontal translationHorizontal translation(phase shift)(phase shift)

Vertical translationVertical translation

Careful, Careful, Careful !!Careful, Careful, Careful !!

f(x – 1) is f(x) shifted to the f(x – 1) is f(x) shifted to the RightRight by 1 by 1

Let’s say we start with:Let’s say we start with: )2sin()( xxf

Then: f(x – 1) = sin(2(x -1))Then: f(x – 1) = sin(2(x -1)) (Replace ‘x’ with ‘x – 1’) (Replace ‘x’ with ‘x – 1’)

Distributive property: f(x – 1) = sin(2x – 2)

IMPORTANTIMPORTANT: horizontal shifts are ‘: horizontal shifts are ‘c ÷ bc ÷ b’ !!!!’ !!!!

Horizontal stretches affect Horizontal stretches affect bothboth the variable the variable along the x-axis along the x-axis ANDAND the phase shift. the phase shift.

dcbxaxf sin)(

532sin3)( xxf

dcxbaxf )(sin)(Vertical stretch/shrink: Vertical stretch/shrink: factor of 3factor of 3

If a < 0: reflection across x-axis: If a < 0: reflection across x-axis: nonenone

Horizontal stretch/shrink by factor of: Horizontal stretch/shrink by factor of: 2

1

532sin3)( xxf

dcbxaxf sin)(Vertical stretch/shrink: Vertical stretch/shrink: factor of 3factor of 3

If a < 0: reflection across x-axis: If a < 0: reflection across x-axis: nonenone

Horizontal stretch/shrink by factor of: Horizontal stretch/shrink by factor of: 2

1

If b < 0: reflection across y-axis: If b < 0: reflection across y-axis: nonenone

Horizontal translation: Horizontal translation: left byleft by

Vertical translation: Vertical translation: up by 5 unitsup by 5 units

623 b

c

Your turn: Your turn: describe the transformations of f(x)describe the transformations of f(x)

2433sin5.0)( xxf

8. Vertical stretch/shrink8. Vertical stretch/shrink9. Horizontal stretch/shrink9. Horizontal stretch/shrink10. Horizontal translation (phase shift)10. Horizontal translation (phase shift)11. Vertical translation 11. Vertical translation

SinusoidSinusoid

1.1. AmplitudeAmplitude: : ((½ of peak to peak distance) = ½ of peak to peak distance) =

2. 2. PeriodPeriod: length of horizontal axis encompassing: length of horizontal axis encompassing oneone complete cycle = complete cycle =

a

b

2

3. 3. FrequencyFrequency = = 2

b

dcbxaxf )sin()(

Frequency = 1/periodFrequency = 1/period

532sin3)( xxf

dcbxaxf sin)(1.1. AmplitudeAmplitude: : ((½ of peak to peak distance) = ½ of peak to peak distance) =

2. 2. PeriodPeriod: length of horizontal axis encompassing: length of horizontal axis encompassing oneone complete cycle = complete cycle =

33

2

2

3. 3. FrequencyFrequency = = 1

Your turn: Your turn: describe the transformations of f(x)describe the transformations of f(x)

2433sin5.0)( xxf

12. Amplitude = ?12. Amplitude = ?13. Period = ?13. Period = ?14. Frequency = ?14. Frequency = ?

Sine vs. Cosine Sine vs. Cosine Function Function

Cos xCos x = = sin xsin x shifted to the left by shifted to the left by 2

2sincos xx

Construct a SinusoidConstruct a Sinusoid dcbxaxf sin)(

Period = Period = 5

Amplitude = 6Amplitude = 6 Passes thru: (2, 0)Passes thru: (2, 0)

SolutionSolution: : 1. Find ‘b’ (coefficient of ‘x’)1. Find ‘b’ (coefficient of ‘x’) period

b

2

b 2

5 10b

2. Amplitude is easy: a = ± 62. Amplitude is easy: a = ± 6

Either works, use +10Either works, use +10



Period = Period = 5

Amplitude = 6Amplitude = 6 Passes thru: (2, 0)Passes thru: (2, 0)

3. Find ‘c’: does everything 3. Find ‘c’: does everything butbut pass through (2, 0). pass through (2, 0).

It It doesdoes pass through (0, 0). We just need a phase pass through (0, 0). We just need a phase shift of +2.shift of +2.

xxf 10sin6)(

2010sin6)( xxf

)2(10sin6)2( xxf

After distributive property: After distributive property:


Period = Period = 3

2 Amplitude = 2Amplitude = 2 Passes thru: (0, 7)Passes thru: (0, 7)


b

2

b 2

32 3b





Period = Period = 3



It It doesdoes pass through (0, 0). We just need to shift pass through (0, 0). We just need to shift up by 7. up by 7.

xxf 3sin2)(

73sin2)( xxf


Period = Period = 9



b

2

b 2

94 2

9b


Either works, useEither works, use


29b


Period = Period = 9



It It doesdoes pass through (0, 0). We just need to shift pass through (0, 0). We just need to shift up by 7 up by 7 andand have a phase have a phase shift of 2 to the right.shift of 2 to the right.

xxf 29sin4)(

7929sin4)( xxf

7)2(29sin4)2( xxf

After distributive property: After distributive property:

HOMEWORKHOMEWORK

Section 4-4Section 4-4

Another exampleAnother examplef(x) = ? When: f(x) = ? When: minimum value: y = 5 at x = 0minimum value: y = 5 at x = 0 and maximum value: y = 25 at x = 32and maximum value: y = 25 at x = 32

1. 1. Find amplitudeFind amplitude: :

Remember: the amplitude is Remember: the amplitude is halfhalf the “peak to peak” distancethe “peak to peak” distance

102

525

a

2. 2. Find periodFind period::

Period is the horizontal distance to fromPeriod is the horizontal distance to from minimum to maximum and minimum to maximum and back to minimumback to minimum. .

Period = 64Period = 64b

264

32

b


4. Use a horizontal translation of either the 4. Use a horizontal translation of either the sine or cosine function to match up (0, 5)sine or cosine function to match up (0, 5)

Cosine has a Cosine has a maximum valuemaximum value at x = 0 at x = 0 Reflect cosine Reflect cosine across x-axis to get a across x-axis to get a minimumminimum value at x = 0. value at x = 0.

xxf

32cos10)(

5. Vertical translation to make5. Vertical translation to make minimum value = 5minimum value = 5

Move up 15 units.Move up 15 units.15

32cos10)(

xxf


1. 1. Find amplitudeFind amplitude: : 10a

2. 2. Find periodFind period:: Period = 64Period = 64

3. 3. Find the coefficient of ‘x’Find the coefficient of ‘x’:: 32

b

4. Use a horizontal translation of either the 4. Use a horizontal translation of either the sine or cosine function to match up (0, 5)sine or cosine function to match up (0, 5)

5. Vertical translation to make5. Vertical translation to make minimum value = 5minimum value = 5

Quiz 4-3 1.Find a positive and negative co-terminal angle with: co-terminal angle with: 2.Find a positive and negative co-terminal angle with: co-terminal.

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