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### Feb 11, 2011

The transformed trigonometric functions

f(x) = a sin b(x – h) + k

- Recall which is which in the rule:

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 amplitude: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:

Amplitude = half the distance between the Max and min values

= (M – m) 2

= (2 - -6) 2

= 8 2

= 4

2

-6

What would be the value of a in the rule?

a = 1

Amplitude = half the distance 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:

- For f(x) = a sin b(x – h) + k
OR:

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

In general then:

- For f(x) = a sin b(x – h) + k
OR:

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

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 frequency:In general then:

- For f(x) = a sin b(x – h) + k
OR:

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

In general then:

- For f(x) = a sin b(x – h) + k
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...

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.

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.

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 = 8 sin 2x

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

y = 2.4 sin (-9x) - 2

period =

period =

period =

period =

What would be the period:In general then:

- For f(x) = a sin b(x – h) + k
OR:

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

Period =

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

Amplitude

Period

Frequency

l.o.o.

y = cos 4x + 1 name:

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 name:

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 =

What would be the phase shift:What would be the value of h in the rule? name:

If we consider this to be a sine function,

h =

Snake is beginning here

What would be the value of h in the rule? name:

If we consider this to be a sine function,

h =

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

What would be the value of h in the rule? name:

If we consider this to be a sine function,

h =

In the rule, you would see:

What would be the value of h in the rule? name:

Tulip is beginning here

If we consider this to be a cos function,

h =

What would be the value of h in the rule? name:

Which is to the right of where it usually begins

If we consider this to be a cos function,

h =

What would be the value of h in the rule? name:

Which is to the right of where it usually begins

If we consider this to be a cos function,

h =

What would be the value of h in the rule? name:

If we consider this to be a cos function,

h =

In the rule, you would see:

(x - )

If considered as a sine function, name:h =

If considered as a cos function, name:h =

y = cos 4x + 1 name:

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 name:

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 l.o.o.:What would be the value of k in the rule? name:

k = -1

Another way to find k: name:

k = the number halfway between the Max and min values

= (M + m) 2

= (1 + -3) 2

= -2 2

= -1

Another way to find k: name:

k = the number halfway between the Max and min values

= (M + m) 2

= (1 + -3) 2

= -2 2

= -1

What would be the value of k in the rule? name:

k = the number halfway between the Max and min values

= (M + m) 2

= (0 + -2) 2

= -2 2

= -1

In general then: name:

- 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

And another thing.... name:

- For f(x) = a sin b(x – h) + k
OR:

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

min = k - amplitude

And another thing.... name:

- 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 = 3 sin 2x - 1 name:

y name:= 3 sin 2x- 1

y = -1

y = name:3 sin 2x - 1

y = -1

y = name:3 sin 2x - 1

2

y = name:3 sin 2x - 1

2

y name:= 3 sin 2x - 1

P = 2/2 =

Find the rule: name:

y = 2 cos x name:

Find the rule: name:

y = 3 sin x name:

Find the rule: name:

y = 3 sin 2x name:

Find the rule: name:

y = 3 sin 2x - 1 name:

Find the rule: name:

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

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

Hwk: name:

- 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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