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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

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:




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:







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 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:








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.

?


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.


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 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:


What would be the phase shift1

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 rule1
What would be the value of h in the rule? name:

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? name:

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? 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 rule4
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 rule5
What would be the value of h in the rule? name:

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? name:

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? 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 rule8
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 rule9
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 - )







Which is affected by parameter k
Which is affected by parameter k? name:

k = 0

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter k1
Which is affected by parameter k? name:

k = 1

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter k2
Which is affected by parameter k? name:

k = 2

Amplitude

Period

Frequency

l.o.o.


Which is affected by parameter k3
Which is affected by parameter k? name:

Amplitude

Period

Frequency

l.o.o.


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

Amplitude

Period

Frequency

l.o.o.


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 + )

What would be the l.o.o.:


What would be the l o o1

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.:




Another way to find k
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 k1
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 rule3
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 then6
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
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 thing1
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 11
y name:= 3 sin 2x- 1

y = -1


Y 3 sin 2 x 1
y = name:3 sin 2x - 1

y = -1


Y 3 sin 2 x 11
y = name:3 sin 2x - 1

2


Y 3 sin 2 x 12
y = name:3 sin 2x - 1

2


Y 3 sin 2 x 13
y name:= 3 sin 2x - 1

P = 2/2 = 











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


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


Hwk: name:

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