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

Feb 11, 2011

The transformed trigonometric functions


F x a sin b x h k

f(x) = a sin b(x – h) + k

  • Recall which is which in the rule:


Match the parameters to the number

Match the parameters to the number:

k

h

b

a


Match the parameters to the number1

Match the parameters to the number:

k

h

b

a

5

7

4

1


Which is affected by parameter a

Which is affected by parameter a?

a = 1

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter a1

Which is affected by parameter a?

a = 2

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter a2

Which is affected by parameter a?

a = 3

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter a3

Which is affected by parameter a?

Amplitude

Period

Frequency

l.o.o.


In fact parameter a amplitude

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:


What would be the amplitude1

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?


What would be the value of a in the rule1

What would be the value of a in the rule?

a = 5


What would be the value of a in the rule2

What would be the value of a in the rule?


What would be the value of a in the rule3

What would be the value of a in the rule?

a = 4


What would be the value of a in the rule4

What would be the value of a in the rule?

a = 4


Another way to find 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 amplitude1

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 rule5

What would be the value of a in the rule?


What would be the value of a in the rule6

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

In general then:

  • For f(x) = a sin b(x – h) + k

    OR:

  • f(x) = a cos b(x – h) + k

    Amplitude =


In general then1

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 then2

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

Which is affected by parameter b?

b = 1

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter b1

Which is affected by parameter b?

b = 2

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter b2

Which is affected by parameter b?

b = 4

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter b3

Which is affected by parameter b?

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter b4

Which is affected by parameter b?

4 cycles

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter b5

Which is affected by parameter b?

Amplitude

Period

Frequency

l.o.o.


In fact b frequency

In fact, b = frequency

y = sin 4x

Amplitude

Period

Frequency = 4 = b

l.o.o.


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:


What would be the frequency1

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?


What would be the value of b in the rule1

What would be the value of b in the rule?

b = 1


What would be the value of b in the rule2

What would be the value of b in the rule?


What would be the value of b in the rule3

What would be the value of b in the rule?

b = 3


What would be the value of b in the rule4

What would be the value of b in the rule?


What would be the value of b in the rule5

What would be the value of b in the rule?

b = 0.5


In general then3

In general then:

  • For f(x) = a sin b(x – h) + k

    OR:

  • f(x) = a cos b(x – h) + k

    Frequency =


In general then4

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

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 of1

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.

?


Feb 11 2011

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.


Feb 11 2011

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.


Feb 11 2011

In fact, period =

y = sin 4x

Amplitude

Period =

Frequency

l.o.o.


Feb 11 2011

In fact, period =

y = sin 4x

Amplitude

Period =

Frequency

l.o.o.


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:


What would be the period1

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 then5

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

Which is affected by parameter h?

h = 0

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter h1

Which is affected by parameter h?

h = .3

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter h2

Which is affected by parameter h?

h = .5

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter h3

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

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:


What would be the phase shift1

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?


What would be the value of h in the rule1

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 rule2

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 rule3

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 rule4

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 rule5

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 rule6

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 rule7

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 rule8

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 rule9

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 rule10

What would be the value of h in the rule?


If considered as a sine function h

If considered as a sine function,h =


If considered as a cos function h

If considered as a cos function,h =


What would be the value of h in the rule11

What would be the value of h in the rule?


What would be the value of h in the rule12

What would be the value of h in the rule?

As a cos:

h = 0


Which is affected by parameter k

Which is affected by parameter k?

k = 0

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter k1

Which is affected by parameter k?

k = 1

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter k2

Which is affected by parameter k?

k = 2

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter k3

Which is affected by parameter k?

Amplitude

Period

Frequency

l.o.o.


In fact l o o has equation y k

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.:


What would be the l o o1

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?


What would be the value of k in the rule1

What would be the value of k in the rule?

k = -1


Another way to find k

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 k1

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 rule2

What would be the value of k in the rule?


What would be the value of k in the rule3

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 then6

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

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 thing1

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 3 sin 2x 11

y = 3 sin 2x- 1

y = -1


Y 3 sin 2 x 1

y = 3 sin 2x - 1

y = -1


Y 3 sin 2 x 11

y = 3 sin 2x - 1

2


Y 3 sin 2 x 12

y = 3 sin 2x - 1

2


Y 3 sin 2 x 13

y = 3 sin 2x - 1

P = 2/2 = 


Find the rule

Find the rule:


Y 2 cos x

y = 2 cos x


Find the rule1

Find the rule:


Y 3 sin x

y = 3 sin x


Find the rule2

Find the rule:


Y 3 sin 2x

y = 3 sin 2x


Find the rule3

Find the rule:


Y 3 sin 2x 12

y = 3 sin 2x - 1


Find the rule4

Find the rule:


Y 2 sin 3 x 4 1

y = 2 sin 3(x - /4) + 1


Y 2 cos 3 x 4 1

y = 2 cos 3(x + /4) + 1


Feb 11 2011

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