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 UNIT 2 : FUNCTIONS

 LESSON 6: STRETCHES & COMPRESSIONS

 

1. Vertical Stretches and Compressions:

 

Example 1: Given the graph of y = f(x) as shown [blue], draw the graphs of y = 4f(x), and y = f(x)

 


 


Solution:

 

The required graph of y = 4f(x) will be a vertical stretch factor 4 and may be represented in mapping form:

 

(x, y) ---------------------------(x, 4y)

 

 
 

 

 


(Now take key points on the graph of y = f(x) -- {(-5, 1), (-3, 1), (-1, 3),(0,1.5), (1, 0), (3, 2)} and multiply the y-coordinates by 4.

 

(x, y) ----------------------------(x, 4y)

(-5, 1) ---------------------------(-5, 4)

(-3, 1) ---------------------------(-3, 4)

(-1, 3) ---------------------------(-1, 12)

(1, 0) ----------------------------(1, 0)

(3, 2) ----------------------------(3, 8) [see red graph at left]

 

 

 

 

 

 

 

 

The required graph of y = f(x) will be a vertical compression factor and may be represented in mapping form:

(x, y) ---------------------------(x, y)

 

 
 

 

 

 


(Now take key points on the graph of y = f(x) -- {(-5, 1), (-3, 1), (-1, 3),(0,1.5), (1, 0), (3, 2)} and multiply the y-coordinates by or 0.5

 

(x, y) ----------------------------(x, y) or (x, 0.5y)

(-5, 1) ---------------------------(-5, 0.5)

(-3, 1) ---------------------------(-3, 0.5)

(-1, 3) ---------------------------(-1, 1.5)

(1, 0) ----------------------------(1, 0)

(3, 2) ----------------------------(3, 1) [see green graph above left]

 

 

Example 2: Given the graph of y = x2 as shown, draw the graphs of y = 3f(x), and y = f(x)

 

 

Solution:

 

The required graph of y = 3f(x) will be a vertical stretch factor 3 and may be represented in mapping form:

(x, y) ---------------------------(x, 3y)

 

 
 

 

 

 


(Now take key points on the graph of y = x2 -- {(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)} and multiply the y-coordinates by 3.

 

 

(x, y) ----------------------------(x, 3y)

(-2, 4) ---------------------------(-2, 12)

(-1, 1) ---------------------------(-1, 3)

(0, 0) ----------------------------(0, 0)

(1, 1) ----------------------------(1, 3)

(2, 4) ----------------------------(2, 12) [see red graph y = 3x2 at left]

 

 

 

 

 

 

 

 

 

 

 

The required graph of y = f(x) will be a vertical compression factor and may be represented in mapping form:

(x, y) ---------------------------(x, y)

 

 
 

 

 

 


(Now take key points on the graph of y = x2 -- {(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)} and multiply the y-coordinates by or 0.5

 

(x, y) -----------------------------(x, 3y)

(-2, 4) ---------------------------(-2, 2)

(-1, 1) ---------------------------(-1, 0.5)

(0, 0) ----------------------------(0, 0)

(1, 1) ----------------------------(1, 0.5)

(2, 4) ----------------------------(2, 2) [see green graph y = x2 above left]

 

.

In summary, if k is an integer:

The graph of kf(x) will be a vertical stretch of factor k

and the graph of 1/k f(x) will be a vertical compression of factor 1/k

 
 

 

 

 

 

 

 


2. Horizontal Stretches and Compressions:

 

Example 3: Given the graph of y = f(x) as shown [blue], draw the graphs of y = f(2x), and y = f(1/2 x)

 


 

 


Solution: For f(2x), complete the table of values below; when x = -2.5, 2x will be 5 and read the

value for f(-5) from the blue graph. Here f(-5) = 1.

 

x

2x

f(2x)

-2.5

-5

1

-2

-4

1

-1.5

-3

1

-1

-2

2

-0.5

-1

3

0.5

1

0

1

2

1

1.5

3

2

 

 

 

 

 

 

 

Yielding the ordered pairs for f(2x) : {(-2.5, 1), (-2, 1), (-1.5, 1), (-1, 2), (-0.5, 3), (0.5, 0), (1, 1) (1.5, 2)} and graph above [red]

Note that the required graph of y = f(2x) [red] will be a horizontal compression factor of the graph of f(x) [blue]

and may be represented in mapping form:

 

(x, y) ------------------------- (1/2 x, y) or (0.5x, y)

 

 
 

 

 

 

 


For f(1/2 x), complete the table of values below;

 

 

x

x

f( x)

-10

-5

1

-6

-3

1

-2

-1

3

2

1

0

4

2

1

6

3

2

 

 

 

 

 

 

 

 

Yielding the ordered pairs for f( x) : {(-10, 1), (-6, 1), (-2, 3), (2, 0), (4, 1), (6, 2)} and graph above [green]

Note that the required graph of y = f( x) [green] will be a horizontal stretch factor 2 of the graph of f(x) [blue]

and may be represented in mapping form:

 

 

(x, y) ------------------------- (2x, y)

 

 
 

 

 

 

 

In summary, if k is an integer:

The graph of f(kx) will be a horizontal compression of factor 1/k

and the graph of f(1/k x) will be a horizontal stretch of factor k

 
 

 

 

 

 

 

 

 


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