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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:. Match the parameters to the number:. k. h. b. a. Match the parameters to the number:. k. h. b. a. 5. 7. 4. 1. Which is affected by parameter a?. a = 1.

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Feb 11, 2011

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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:

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

### 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

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:

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|

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.

### In fact, b = frequency

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

### 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

b = 1

b = 3

b = 0.5

### 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|

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.

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 =

### 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 =

### In general then:

• 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.

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

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 =

### 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 =

### What would be the value of h in the rule?

If we consider this to be a sine function,

h =

### What would be the value of h in the rule?

If we consider this to be a sine function,

h =

Snake is beginning here

### What would be the value of h in the rule?

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?

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?

If we consider this to be a cos function,

h =

### What would be the value of h in the rule?

Tulip is beginning here

If we consider this to be a cos function,

h =

### What would be the value of h in the rule?

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?

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?

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.

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

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 + )

### 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

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:

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?

k = the number halfway between the Max and min values

= (M + m)  2

= (0 + -2)  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

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

### And another thing....

• 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....

• 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 = 

### 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