Feb 11, 2011

Feb 11, 2011

The transformed trigonometric functions

f(x) = a sin b(x – h) + k

• Recall which is which in the rule:

Match the parameters to the number:

f(x) 5 sin4(x 1) 7

a b h k

Match the parameters to the number:

f(x) 5 sin4(x 1) 7

a b h k

5 74 1

Which is affected by parameter a?

Amplitude

Period

Frequency

l.o.o.

a = 1


Amplitude

Period

Frequency

l.o.o.

a = 2


Amplitude

Period

Frequency

l.o.o.

a = 3


Amplitude

Period

Frequency

l.o.o.

In fact, parameter a = amplitude

Amplitude

Period

Frequency

l.o.o.

What would be the amplitude:

• y = 2 cos x

• y = 8 sin 2x

• y = -3 cos x

• y = 4 sin 9x - 2

What would be the amplitude:

• y = 2 cos x

• y = 8 sin 2x

• y = -3 cos x

• y = 2.4 sin 9x - 2

• amplitude = 2

• amplitude = 8

• amplitude = 3

• amplitude = 2.4

What would be the value of a in the rule?


a = 5



a = 4


a = 4

Another way to find amplitude:

Amplitude = half the distance

between the Max and min values= (M – m) 2= (2 - -6) 2

= 8 2= 4

Another way to find amplitude:


between the Max and min values= (M – m) 2= (2 - -6) 2

= 8 2= 4

2

-6



a = 1


between the Max and min values= (M – m) 2= (2 - 0) 2

= 2 2= 1

In general then:

• For f(x) = a sin b(x – h) + k

OR:

• f(x) = a cos b(x – h) + k

Amplitude =

In general then:


OR:

• f(x) = a cos b(x – h) + k

Amplitude = |a|

In general then:


OR:

• f(x) = a cos b(x – h) + k

Amplitude = |a|Max min

2

Which is affected by parameter b?

Amplitude

Period

Frequency

l.o.o.

b = 1


Amplitude

Period

Frequency

l.o.o.

b = 2


Amplitude

Period

Frequency

l.o.o.

b = 4


Amplitude

Period

Frequency

l.o.o.


Amplitude

Period

Frequency

l.o.o.

4 cycles


Amplitude

Period

Frequency

l.o.o.

In fact, b = frequency

Amplitude

Period

Frequency = 4 = b

l.o.o.

y = sin 4x

What would be the frequency:

• y = cos 4x

• y = 8 sin 2x

• y = -3 cos (x + 1) -2

• y = 2.4 sin (-9x) - 2

What would be the frequency:

• y = cos 4x

• y = 8 sin 2x

• y = -3 cos (x + 1) -2

• y = 2.4 sin (-9x) - 2

• frequency = 4

• frequency = 2

• frequency =

• frequency = 9

What would be the value of b in the rule?


b = 1



b = 3



b = 0.5

In general then:


OR:

• f(x) = a cos b(x – h) + k

Frequency =

In general then:


OR:

• f(x) = a cos b(x – h) + k

Frequency = |b|

And if 4 cycles have a total width of 2.... ...then one of those cycles must have

a width of...

Amplitude

Period

Frequency

l.o.o.

y = sin 4x


a width of...

Amplitude

Period

Frequency

l.o.o.

y = sin 4x

?

Amplitude

Period =

Frequency

l.o.o.

y = sin 4x

2

4


a width of...

Amplitude

Period =

Frequency

l.o.o.

y = sin 4x

2

4

2


a width of...

Amplitude

Period =

Frequency

l.o.o.

y = sin 4x

2

4

In fact, period =

2

Amplitude

Period =

Frequency

l.o.o.

y = sin 4x

2

4

In fact, period =

2

2

b

What would be the period:

• y = cos 4x

• y = 8 sin 2x

• y = -3 cos (x + 1) -2

• y = 2.4 sin (-9x) - 2

• period =

• period =

• period =

• period =

What would be the period:

• y = cos 4x

• y = 8 sin 2x

• y = -3 cos (x + 1) -2

• y = 2.4 sin (-9x) - 2

• period =

• period =

• period =

• period =

2

4 2

2

2

22

2

9

In general then:


OR:

• f(x) = a cos b(x – h) + k

Frequency = |b|

Period = 2

b

Which is affected by parameter h?

Amplitude

Period

Frequency

l.o.o.

h = 0


Amplitude

Period

Frequency

l.o.o.

h = .3


Amplitude

Period

Frequency

l.o.o.

h = .5


Amplitude

Period

Frequency

l.o.o.

But h does shift horizontally...and this shift has a special name:

Phase shift

Amplitude

Period

Frequency

l.o.o.

What would be the phase shift:

• y = cos 4x + 1

• y = 8 sin 2(x - ) -3

• y = -3 cos (x + 1) -2

• y = 2.4 sin (2x + )

• phase shift =

• phase shift =

• phase shift =

• phase shift =

What would be the phase shift:

• y = cos 4x + 1

• y = 8 sin 2(x - ) -3

• y = -3 cos (x + 1) -2

• y = 2.4 sin (2x + )

• phase shift = 0

• phase shift =

• phase shift = -1

• phase shift = 2

What would be the value of h in the rule?


If we consider this to be a sine function,

h = 2



h = 2

Snake is beginning

here



h = 2

Which is /2 to the right of

where it usually begins



h =

In the rule, you would see:

2

x2


If we consider this to be a cos function,

h =



h =

Tulip is beginning

here



h =

Which is to the right of




h =

Which is to the right of




h =

In the rule, you would see:

(x - )


If considered as a sine function,h =

4

If considered as a cos function,h = 3

4



As a cos:h = 0

Which is affected by parameter k?

Amplitude

Period

Frequency

l.o.o.

k = 0


Amplitude

Period

Frequency

l.o.o.

k = 1


Amplitude

Period

Frequency

l.o.o.

k = 2


Amplitude

Period

Frequency

l.o.o.

In fact, l.o.o. has equation: y = k

Amplitude

Period

Frequency

l.o.o.

What would be the l.o.o.:

• y = cos 4x + 1

• y = 8 sin 2(x - ) - 3

• y = -3 cos (x + 1) - 2

• y = 2.4 sin (2x + )

What would be the l.o.o.:

• y = cos 4x + 1

• y = 8 sin 2(x - ) - 3

• y = -3 cos (x + 1) - 2

• y = 2.4 sin (2x + )

• l.o.o.: y = 1

• l.o.o.: y = -3

• l.o.o.: y = -2

• l.o.o.: y = 0

What would be the value of k in the rule?


k = -1

Another way to find k:

k = the number halfway between the Max and min

values= (M + m) 2= (1 + -3) 2

= -2 2= -1

Another way to find k:


values= (M + m) 2= (1 + -3) 2

= -2 2= -1




values= (M + m) 2= (0 + -2) 2

= -2 2= -1

In general then:


OR:

• f(x) = a cos b(x – h) + k

l.o.o. is the line y = k

Max mink

2

And another thing....


OR:

• f(x) = a cos b(x – h) + k

Max = k + amplitudemin = k - amplitude

And another thing....


OR:

• f(x) = a cos b(x – h) + k

Max = k + amplitudemin = k - amplitude

y = 3 sin 2x - 1

y = 3 sin 2x - 1

y = -1

y = 3 sin 2x - 1

y = -1

y = 3 sin 2x - 1

2

y = 3 sin 2x - 1

2

y = 3 sin 2x - 1

P = 2/2 =

Find the rule:

y = 2 cos x

Find the rule:

y = 3 sin x

Find the rule:

y = 3 sin 2x

Find the rule:

y = 3 sin 2x - 1

Find the rule:y

x

2

2

32

32

2

2

–2

–2

–

–

–32

–32

– 2

– 2

1

1

2

2

3

3

4

4

5

5

– 1

– 1

– 2

– 2

y = 2 sin 3(x - /4) + 1y

x

2

2

32

32

2

2

–2

–2

–

–

–32

–32

– 2

– 2

1

1

2

2

3

3

4

4

5

5

– 1

– 1

– 2

– 2

y = 2 cos 3(x + /4) + 1y

x

2

2

32

32

2

2

–2

–2

–

–

–32

–32

– 2

– 2

1

1

2

2

3

3

4

4

5

5

– 1

– 1

– 2

– 2

Hwk:

• Blog • Three gizmos:

– Cosine function– Sine function– Translating and scaling Sine and Cosine

functions – Activity A

• Carousel:– p. 253 #6, 9ab, 10abd, 19– p. 263 #6, 9, 10

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Feb 11, 2011