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.

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