College Algebra, 2013a

130 Relations and Functions
10
y
10
y
(5, 10)
9
9
8
8
7
6
5
(5, 5)
7
6
5
(2, 6)
(4, 6)
4
3
2
(2, 3)
(4, 3)
4
3
(0, 2)
(0, 1)
1
1 2 3 4 5
y = f(x)
x
vertical scaling by a factor of 2
−−−−−−−−−−−−−−−−−−−−→
multiply each y-coordinate by 2
1 2 3 4 5
y =2f(x)
x
If we wish to graph y = 1 2f(x), we multiply the all of the y-coordinates of the points on the graph
of f by 1 2 . This creates a ‘vertical scaling8 by a factor of 1 2
’ as seen below.
5
y
(5, 5)
5
y
4
3
2
(0, 1)
(2, 3)
(4, 3)
1 2 3 4 5
y = f(x)
x
vertical scaling by a factor of 1 2
−−−−−−−−−−−−−−−−−−−−→
multiply each y-coordinate by 1 2
4
3
2
( ) 1
0,
1
2
( )
2,
3
2
( )
4,
3
2
( )
5,
5
2
1 2 3 4 5
y = 1 2 f(x)
x
These results are generalized in the following theorem.
Theorem 1.5. Vertical Scalings. Suppose f is a function and a>0. To graph y = af(x),
multiply all of the y-coordinates of the points on the graph of f by a. Wesaythegraphoff
has been vertically scaled by a factor of a.
ˆ If a>1, we say the graph of f has undergone a vertical stretching (expansion, dilation)
by a factor of a.
ˆ If 0