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Feb 11, 2011

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Feb 11, 2011

The transformed trigonometric functions

- Recall which is which in the rule:

k

h

b

a

k

h

b

a

5

7

4

1

a = 1

Amplitude

Period

Frequency

l.o.o.

a = 2

Amplitude

Period

Frequency

l.o.o.

a = 3

Amplitude

Period

Frequency

l.o.o.

Amplitude

Period

Frequency

l.o.o.

Amplitude

Period

Frequency

l.o.o.

y = 2 cos x

y = 8 sin 2x

y = -3 cos x

y = 4 sin 9x - 2

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

a = 5

a = 4

a = 4

Amplitude = half the distance between the Max and min values

= (M – m) 2

= (2 - -6) 2

= 8 2

= 4

Amplitude = half the distance between the Max and min values

= (M – m) 2

= (2 - -6) 2

= 8 2

= 4

2

-6

a = 1

Amplitude = half the distance between the Max and min values

= (M – m) 2

= (2 - 0) 2

= 2 2

= 1

- For f(x) = a sin b(x – h) + k
OR:

- f(x) = a cos b(x – h) + k
Amplitude =

- For f(x) = a sin b(x – h) + k
OR:

- f(x) = a cos b(x – h) + k
Amplitude = |a|

- For f(x) = a sin b(x – h) + k
OR:

- f(x) = a cos b(x – h) + k
Amplitude = |a|

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.

Amplitude

Period

Frequency

l.o.o.

y = sin 4x

Amplitude

Period

Frequency = 4 = b

l.o.o.

y = cos 4x

y = 8 sin 2x

y = -3 cos (x + 1) -2

y = 2.4 sin (-9x) - 2

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

b = 1

b = 3

b = 0.5

- For f(x) = a sin b(x – h) + k
OR:

- f(x) = a cos b(x – h) + k
Frequency =

- For f(x) = a sin b(x – h) + k
OR:

- f(x) = a cos b(x – h) + k
Frequency = |b|

y = sin 4x

Amplitude

Period

Frequency

l.o.o.

y = sin 4x

Amplitude

Period

Frequency

l.o.o.

?

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

y = sin 4x

Amplitude

Period =

Frequency

l.o.o.

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

y = sin 4x

Amplitude

Period =

Frequency

l.o.o.

In fact, period =

y = sin 4x

Amplitude

Period =

Frequency

l.o.o.

In fact, period =

y = sin 4x

Amplitude

Period =

Frequency

l.o.o.

y = cos 4x

y = 8 sin 2x

y = -3 cos (x + 1) -2

y = 2.4 sin (-9x) - 2

period =

period =

period =

period =

y = cos 4x

y = 8 sin 2x

y = -3 cos (x + 1) -2

y = 2.4 sin (-9x) - 2

period =

period =

period =

period =

- For f(x) = a sin b(x – h) + k
OR:

- f(x) = a cos b(x – h) + k
Frequency = |b|

Period =

h = 0

Amplitude

Period

Frequency

l.o.o.

h = .3

Amplitude

Period

Frequency

l.o.o.

h = .5

Amplitude

Period

Frequency

l.o.o.

Amplitude

Period

Frequency

l.o.o.

Amplitude

Period

Frequency

l.o.o.

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 =

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 =

If we consider this to be a sine function,

h =

If we consider this to be a sine function,

h =

Snake is beginning here

If we consider this to be a sine function,

h =

Which is /2 to the right of where it usually begins

If we consider this to be a sine function,

h =

In the rule, you would see:

If we consider this to be a cos function,

h =

Tulip is beginning here

If we consider this to be a cos function,

h =

Which is to the right of where it usually begins

If we consider this to be a cos function,

h =

Which is to the right of where it usually begins

If we consider this to be a cos function,

h =

If we consider this to be a cos function,

h =

In the rule, you would see:

(x - )

As a cos:

h = 0

k = 0

Amplitude

Period

Frequency

l.o.o.

k = 1

Amplitude

Period

Frequency

l.o.o.

k = 2

Amplitude

Period

Frequency

l.o.o.

Amplitude

Period

Frequency

l.o.o.

Amplitude

Period

Frequency

l.o.o.

y = cos 4x + 1

y = 8 sin 2(x - ) - 3

y = -3 cos (x + 1) - 2

y = 2.4 sin (2x + )

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

k = -1

k = the number halfway between the Max and min values

= (M + m) 2

= (1 + -3) 2

= -2 2

= -1

k = the number halfway between the Max and min values

= (M + m) 2

= (1 + -3) 2

= -2 2

= -1

k = the number halfway between the Max and min values

= (M + m) 2

= (0 + -2) 2

= -2 2

= -1

- For f(x) = a sin b(x – h) + k
OR:

- f(x) = a cos b(x – h) + k
l.o.o. is the line y = k

- For f(x) = a sin b(x – h) + k
OR:

- f(x) = a cos b(x – h) + k
Max = k + amplitude

min = k - amplitude

- For f(x) = a sin b(x – h) + k
OR:

- f(x) = a cos b(x – h) + k
Max = k + amplitude

min = k - amplitude

y = -1

y = -1

2

2

P = 2/2 =

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