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
The transformed trigonometric functions
k
h
b
a
k
h
b
a
5
7
4
1
a = 1
Amplitude
Period
Frequency
l.o.o.
a = 2
Amplitude
Period
Frequency
l.o.o.
a = 3
Amplitude
Period
Frequency
l.o.o.
Amplitude
Period
Frequency
l.o.o.
Amplitude
Period
Frequency
l.o.o.
y = 2 cos x
y = 8 sin 2x
y = -3 cos x
y = 4 sin 9x - 2
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
a = 5
a = 4
a = 4
Amplitude = half the distance between the Max and min values
= (M – m) 2
= (2 - -6) 2
= 8 2
= 4
Amplitude = half the distance between the Max and min values
= (M – m) 2
= (2 - -6) 2
= 8 2
= 4
2
-6
a = 1
Amplitude = half the distance between the Max and min values
= (M – m) 2
= (2 - 0) 2
= 2 2
= 1
OR:
Amplitude =
OR:
Amplitude = |a|
OR:
Amplitude = |a|
b = 1
Amplitude
Period
Frequency
l.o.o.
b = 2
Amplitude
Period
Frequency
l.o.o.
b = 4
Amplitude
Period
Frequency
l.o.o.
Amplitude
Period
Frequency
l.o.o.
4 cycles
Amplitude
Period
Frequency
l.o.o.
Amplitude
Period
Frequency
l.o.o.
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
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
b = 1
b = 3
b = 0.5
OR:
Frequency =
OR:
Frequency = |b|
y = sin 4x
Amplitude
Period
Frequency
l.o.o.
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 =
y = cos 4x
y = 8 sin 2x
y = -3 cos (x + 1) -2
y = 2.4 sin (-9x) - 2
period =
period =
period =
period =
OR:
Frequency = |b|
Period =
h = 0
Amplitude
Period
Frequency
l.o.o.
h = .3
Amplitude
Period
Frequency
l.o.o.
h = .5
Amplitude
Period
Frequency
l.o.o.
Amplitude
Period
Frequency
l.o.o.
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 =
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 =
If we consider this to be a sine function,
h =
If we consider this to be a sine function,
h =
Snake is beginning here
If we consider this to be a sine function,
h =
Which is /2 to the right of where it usually begins
If we consider this to be a sine function,
h =
In the rule, you would see:
If we consider this to be a cos function,
h =
Tulip is beginning here
If we consider this to be a cos function,
h =
Which is to the right of where it usually begins
If we consider this to be a cos function,
h =
Which is to the right of where it usually begins
If we consider this to be a cos function,
h =
If we consider this to be a cos function,
h =
In the rule, you would see:
(x - )
As a cos:
h = 0
k = 0
Amplitude
Period
Frequency
l.o.o.
k = 1
Amplitude
Period
Frequency
l.o.o.
k = 2
Amplitude
Period
Frequency
l.o.o.
Amplitude
Period
Frequency
l.o.o.
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 + )
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
k = -1
k = the number halfway between the Max and min values
= (M + m) 2
= (1 + -3) 2
= -2 2
= -1
k = the number halfway between the Max and min values
= (M + m) 2
= (1 + -3) 2
= -2 2
= -1
k = the number halfway between the Max and min values
= (M + m) 2
= (0 + -2) 2
= -2 2
= -1
OR:
l.o.o. is the line y = k
OR:
Max = k + amplitude
min = k - amplitude
OR:
Max = k + amplitude
min = k - amplitude
y = -1
y = -1
2
2
P = 2/2 =