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:. 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:


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

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter a?

a = 2

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter a?

a = 3

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter a?

Amplitude

Period

Frequency

l.o.o.


In fact, parameter a = amplitude

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

What would be the amplitude:


What would be the value of a in the rule?


What would be the value of a in the rule?

a = 5


What would be the value of a in the rule?


What would be the value of a in the rule?

a = 4


What would be the value of a in the rule?

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?


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|


Which is affected by parameter b?

b = 1

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter b?

b = 2

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter b?

b = 4

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter b?

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter b?

4 cycles

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter b?

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

What would be the frequency:


What would be the value of b in the rule?


What would be the value of b in the rule?

b = 1


What would be the value of b in the rule?


What would be the value of b in the rule?

b = 3


What would be the value of b in the rule?


What would be the value of b in the rule?

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|


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.


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 =

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 =


Which is affected by parameter h?

h = 0

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter h?

h = .3

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter h?

h = .5

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter h?

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 phase shift:


What would be the value of h in the rule?


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 - )


What would be the value of h in the rule?


If considered as a sine function,h =


If considered as a cos function,h =


What would be the value of h in the rule?


What would be the value of h in the rule?

As a cos:

h = 0


Which is affected by parameter k?

k = 0

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter k?

k = 1

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter k?

k = 2

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter k?

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

What would be the l.o.o.:


What would be the value of k in the rule?


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:

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?


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 = 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 = 2 sin 3(x - /4) + 1


y = 2 cos 3(x + /4) + 1


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