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: a b h k

a b h k

Which is affected by parameter a? Amplitude Period Frequency l.o.o. a = 1

Which is affected by parameter a? Amplitude Period Frequency l.o.o. a = 2

Which is affected by parameter a? Amplitude Period Frequency l.o.o. a = 3

Which is affected by parameter a? Amplitude Period Frequency l.o.o.

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: 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 value of a in the rule?

a = 5

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 =

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? Amplitude Period Frequency l.o.o. b = 1

Which is affected by parameter b? Amplitude Period Frequency l.o.o. b = 2

Which is affected by parameter b? Amplitude Period Frequency l.o.o. b = 4

Which is affected by parameter b? Amplitude Period Frequency l.o.o.

Which is affected by parameter b? Amplitude Period Frequency l.o.o. 4 cycles

Which is affected by parameter b? Amplitude Period Frequency l.o.o.

In fact, b = frequency Amplitude Period Frequency = 4 = b l.o.o. y = sin 4x

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: 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 value of b in the rule?

b = 1

What would be the value of b in the rule?

b = 3

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... Amplitude Period Frequency l.o.o. y = sin 4x

And if 4 cycles have a total width of 2  then one of those cycles must have a width of... Amplitude Period Frequency l.o.o. y = sin 4x ?

Amplitude Period = Frequency l.o.o. y = sin 4x And if 4 cycles have a total width of 2  then one of those cycles must have a width of...

Amplitude Period = Frequency l.o.o. y = sin 4x And if 4 cycles have a total width of 2  then one of those cycles must have a width of...

Amplitude Period = Frequency l.o.o. y = sin 4x In fact, period =

Amplitude Period = Frequency l.o.o. y = sin 4x In fact, 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 =

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 =

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? Amplitude Period Frequency l.o.o. h = 0

Which is affected by parameter h? Amplitude Period Frequency l.o.o. h =.3 

Which is affected by parameter h? Amplitude Period Frequency l.o.o. h =.5 

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.

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 =

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? If we consider this to be a cos function, h = Tulip is beginning here

What would be the value of h in the rule? If we consider this to be a cos function, h = Which is  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 cos function, h = Which is  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 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?

As a cos: h = 0

Which is affected by parameter k? Amplitude Period Frequency l.o.o. k = 0

Which is affected by parameter k? Amplitude Period Frequency l.o.o. k = 1

Which is affected by parameter k? Amplitude Period Frequency l.o.o. k = 2

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.

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.: 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 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?

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 = -1

y = 3 sin 2x - 1 y = -1

y = 3 sin 2x - 1 22

22

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

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