Download Presentation
Feb 11, 2011

Loading in 2 Seconds...

1 / 105

# Feb 11, 2011 - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
Download Presentation

## PowerPoint Slideshow about ' Feb 11, 2011' - camden-marsh

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Feb 11, 2011

The transformed trigonometric functions

f(x) = a sin b(x – h) + k
• Recall which is which in the rule:
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:
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?

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:
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.

Which is affected by parameter h?

h = .3

Amplitude

Period

Frequency

l.o.o.

Which is affected by parameter h?

h = .5

Amplitude

Period

Frequency

l.o.o.

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.

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:
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?

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?

Tulip is beginning here

If we consider this to be a cos function,

h =

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 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 rule?

If we consider this to be a cos function,

h =

In the rule, you would see:

(x - )

Which is affected by parameter k?

k = 0

Amplitude

Period

Frequency

l.o.o.

Which is affected by parameter k?

k = 1

Amplitude

Period

Frequency

l.o.o.

Which is affected by parameter k?

k = 2

Amplitude

Period

Frequency

l.o.o.

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.

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 l.o.o.:
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

P = 2/2 = 

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