CHAPTER 1
LESSON 4
Date
Stretching Graphs of Functions
AW 1.4
MP 1.4
Objective:
 To describe how various stretches (compressions and expansions) of functions
affect graphs and their related equations.
1.Comparing the graphs of y  f(x) and y  af(x)
a) Complete the following tables of values. The first one is completed for you.
y  x2
x y
–3 9
–2 4
–1 1
0 0
1 1
2 4
3 9
y  2x 2
x
–3
–2
–1
0
1
2
3
y
1
y  2x
2
x y
–3
–2
–1
0
1
2
3
b) Use the tables of values to graph and label each of the 3 functions on the grid below.
2
c) How can each of the following graphs be obtained from the graph of y  x ?
2
i) y  2x
1 2
ii) y  2 x
2
2
d) In general, how is the graph of y  ax obtained from the graph of y  x
i) when a  1?
ii) when 0  a  1 ?
Summary:
• If a  1, the graph of y  af (x) is obtained when the graph of y  f (x) undergoes a
_______________
__________________ by a factor of a.
• If 0  a  1, the graph of y  af (x) is obtained when the graph of y  f (x) undergoes
a _______________
__________________
by a factor of a.
Remember that the y–values of y  af (x) are obtained by multiplying each y–value of
y  f (x) by the factor a.

What happens if a  0?
In general, if a  0 , the graph of y  af (x) is obtained when the graph of y  f (x)
undergoes a _____________________________ by a factor of a, along with a
_________________________________________.
Note:
y
 f (x) is often used instead of y  af (x) to emphasize that the
a
parameter a involves a stretch in the y–direction: i.e., a vertical stretch.
The notation

Example 1:
Sketch the graph of y  3 x .
We can obtain the graph of y  3 x from the graph of y  x through two
transformations:
a) _____________________________________________________
b) ______________________________________________________
2.Comparing the graphs of y  f(x) and y  f(ax)
a) Complete the following tables of values. The first one is completed for you.
y x
x
16
9
4
1
0
y
4
3
2
1
0
y  (2x )
x
8
4.5
2
0.5
0
y
y  (0.5x )
x
32
18
8
2
0
y
b) Use the tables of values to graph and label each of the 3 functions on the grid below.
c) How can each of the following graphs be obtained from the graph of y  x ?
i) y  (2x )
ii) y  (0.5x )
d) In general, how is the graph of y  (bx) obtained from the graph of y  x
i) when b 1?
ii) when 0  b 1 ?
Summary:
• If b 1, the graph of y  f (bx) is obtained when the graph of y  f (x) undergoes a
________________
_________________ by a factor of
.
• If 0  b 1, the graph of y  f (bx) is obtained when the graph of y  f (x) undergoes
a
_____________
__________________ by a factor of
.
Notice from your tables that for y  (2x ) to have the same y–values as y  x , the
corresponding x–values of y  (2x ) must be divided by the factor 2.

Thus in general, for y  f (bx) to have the same y–values as y  f (x) , the corresponding
x–values of y  f (bx) must be divided by the factor b.
In other words, if b 1, it takes “less x” to do the job of building the function y  f (bx) ,
so we have a horizontal compression of y  f (x) .

Also, if 0  b 1, it takes “more x” to do the job of building the function y  f (bx) , so
we have a horizontal expansion of y  f (x) .


What happens if b  0?
In general, if b  0, the graph of y  f (bx) is obtained when the graph of y  f (x)
1
undergoes a ________ _______ ____________________ by a factor of b , along with a
__________________________________________.
Example 2:
The grid below contains the graph of a function y  f (x) . Sketch the graph of
1
y  f ( 3 x) .
3. Combined Vertical and Horizontal Stretches
y
The function y  af (bx) [or a  f (bx) ] can be obtained by stretching the function

y  f (x) according to the following rules.

If a  1, there is a vertical expansion by a factor of a.
If 0  a  1, there is a vertical compression by a factor of a.
If a  0 there is also a reflection in the x–axis.
1
If b 1, there is a horizontal compression by a factor of b .
1
If 0  b 1, there is a horizontal expansion by a factor of b .
If b  0, there is also a reflection in the y–axis.
Example 3:
How is the graph of 5y  2x related to the graph of y  x ?
Example 4:
2
The graph of y  9  x is shown below.
2
Sketch the graph of 2y  9  x .
Example 5:
The graph of a function y  f (x) is expanded vertically by a factor of 3, compressed
1
horizontally by a factor of 4 , and then reflected in the y–axis. If the equation of its
image is y  af (bx) , determine the values of a and b for the transformation.
Example 6:
The graph of y  f (x) is shown. Sketch the graph of y  2 f ( 12 x) on the same grid.
Example 7:
1
Describe the transformations whereby the function y  4 f ( 5 x ) is obtained from the
function y  f (x) .
Example 8:
2
The graph of y  9  x is shown below.
Write the equation for each of the following graphs that are transformations of
y  9 x2 .

Example 9:
In each case, write the equation of the mapping that results after the given transformation.
3
A) y  x has been vertically translated 5 units up.
3
B) y  x has been expanded vertically by a factor of 5.
1
3
C) y  x has been compressed vertically by a factor of 5 .
1
3
D) y  x has been compressed horizontally by a factor of 5 .