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The same techniques used to transform the graph of other functions you have studied can
be applied to the graphs of square root functions.
Key Concept
Transformations of Square Root Functions
f
(
x
)
=
a
√
x
-
h
+
k
h
—Horizontal Translation
h
units right if
h
is positive
⎜
h
⎟
units left if
h
is negative
The domain is {
x
|
x
≥
h
}.
k
—Vertical Translation
k
units up if
k
is positive
⎜
k
⎟
units down if
k
is negative
If
a
>
0, then the range is {
f
(
x
) |
f
(
x
)
≥
k
}.
If
a
<
0, then the range is {
f
(
x
) |
f
(
x
)
≤
k
}.
a
—Orientation and Shape
• If
a
<
0, the graph is reflected across the
x
-axis.
• If
⎜
a
⎟
>
1, the graph is stretched vertically.
• If 0
<
⎜
a
⎟
<
1, the graph is compressed vertically.
Graph each function. State the domain and range.
a.
y
=
√
x
-
2
+
5
The minimum point is at (
h
,
k
)
=
(2, 5).
x y
2 5
3 6
4 6.4
5 6.7
6 7
7 7.2
8 7.4
C07-011A-888482
y
x
O
Make a table of values for
x
≥
2, and
graph the function. The graph is the
same shape as
f
(
x
)
=
√
x
, but is
translated 2 units right and 5 units up.
Notice the end behavior. As
x
increases,
y
increases.
D
=
[2,
+∞
), {
x
|
x
≥
2}, or {2
≤
x <
+∞}
R
=
[5,
+∞
), {
y
|
y
≥
5}, or {5
≤
x <
+∞}
b.
y
= -
2
√
x
+
3
-
1
The minimum domain value is at
h
or
-
3. Make a table of values for
x
≥ -
3, and
graph the function. Because
a
is negative, the graph is similar to the graph of
f
(
x
)
=
√
x
, but is reflected in the
x
-axis. Because
⎜
a
⎟
>
1, the graph is vertically
stretched. It is also translated 3 units left and 1 unit down.
x
y
-
3
-
1
-
2
-
3
-
1
-
3.8
0
-
4.5
1
-
5
2
-
5.5
3
-
5.9
C07-012A-888482
y
x
O
D
=
[
-
3,
+∞
), {
x
|
x
≥ -
3},
or {
-
3
≤
x <
+∞}
R =
(
-∞
,
-
1], {
y
|
y
≤ -
1},
or {
-∞ <
x
≤ -
1}
Guided Practice
2A.
f
(
x
)
=
2
√
x
+
4
2B.
f
(
x
)
=
1
_
4
√
x
-
5
+
3
Example 2
Graph Square Root Functions
Follow along with your
graphing calculator as you
watch a
Personal Tutor
graph a square root
function.
Study Tip
Domain and Range
The
limits on the domain and
range also represent the
initial point of the graph of a
square root function.
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