### 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 = ½ 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) ----------------------------à(x, 4y)

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

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

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

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

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

The required graph of y = ½ f(x) will be a vertical compression factor ½  and may be represented in mapping form:

 (x, y) ---------------------------(x, ½ 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 ½ or 0.5

(x, y) ----------------------------à(x, ½ y)  or  (x, 0.5y)

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

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

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

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

(3, 2) ----------------------------à(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 = ½ 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) ----------------------------à(x, 3y)

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

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

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

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

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

The required graph of y = ½ f(x) will be a vertical compression factor ½  and may be represented in mapping form:

 (x, y) ---------------------------(x, ½ 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 ½ or 0.5

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

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

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

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

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

(2, 4) ----------------------------à(2, 2)         [see green graph y = ½ 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 ½  of the graph of f(x) [blue]

and may be represented in mapping form:

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

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

 x ½ x f(½ 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(½ 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(½ 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) -------------------------à (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