HORIZONTAL TRANSFORMATIONS OF FUNCTIONS

 

 

y = cos 3(x - 60)   versus   y = cos (3x - 60)

 

QUESTION A:
How to draw the graph  y = cos 3(x - 60)
from y = cos x
 
QUESTION B:
How to draw the graph  y = cos (3x - 60)
from
y = cos x
SOLUTION A SOLUTION B1
Replace (x) by (3x)  in y = cos x
(Compress the graph horizontally by a factor of 1/3)
Replace (x) by (x - 60)  in y = cos x
(Translate the graph 60o right)
y = cos x    will become  y = cos 3x y = cos x    will become     y = cos (x - 60)
Replace (x) by (x - 60)  in y = cos 3x
(Translate the graph 60o right)
Replace x by 3x   in y = cos (x - 60)
(Compress the graph horizontally by a factor of 1/3)
y = cos 3x    will become    y = cos 3(x - 60) y = cos (x - 60)  will become  y = cos (3x - 60)
SOLUTION B2 (Second Method)
cos (3x - 60) = cos 3(x - 20)

Draw y = cos 3(x - 20) by using same method given at SOLUTION A.
 

 

x

-9

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f (x)

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0

1

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2.5

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1.5

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2.5

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3.5

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3

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1

f (x+1)

-1

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2.5

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1.5

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3.5

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3

2

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x

-9

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f (x)

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2.5

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1.5

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2.5

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3.5

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f (x-2)

 

 

-2

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2.5

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3.5

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x

-9

-8

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-3

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1

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f (x)

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2.5

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3.5

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f (x/2)

 

3

 

2.5

 

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1.5

 

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2.5

 

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x

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

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f (x)

-2

-1

0

1

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2.5

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1.5

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1.5

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2.5

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3.5

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3

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1

f (2x)

 

 

 

 

 

-1

1

3

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1

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Horizontal Transformation: Replace x by (x + 1):  Transfer the graph horizontally 1 unit to the left

 

Horizontal Transformation:  Replace x by (x - 2):  Transfer the graph horizontally 2 unit to the right

 

Horizontal Expansion:  Replace x by x/2:  Expand the graph horizontally by a factor of 2

 

Horizontal Compression:  Replace x by 2x:  Compress the graph horizontally by a factor of 1/2