7.6
Graphing Square Root and
Cube Root Functions
Goal
Your Notes
p Graph square root and cube root functions.
VOCABULARY
Radical function A function that contains a radical
with a variable in its radicand
GRAPHS OF SQUARE ROOT/CUBE ROOT FUNCTIONS
Square Root Function
Cube Root Function
y
y
y5a x
(1, a)
(1, a)
(21, 2a)
}
The domain of y 5 aÏx
is x ≥ 0. The range is
y ≥ 0 when a > 0.
x
(0, 0)
x
(0, 0)
y 5 a3 x
The domain and range
3}
of y 5 a Ï x are
all real numbers .
Graph a Square Root Function
Example 1
}
Graph y 5 2Ïx . Then state its domain and range.
Make a table of values. Then plot the points from the
table and connect them with a smooth curve.
x
0
1
y
0
2
2
2.83
3
4
3.46
4
y
y52 x
1
O
1
x
The radicand of a square root is always nonnegative ,
so the domain is x ≥ 0. The range is y ≥ 0 .
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Your Notes
Graph a Cube Root Function
Example 2
1
2
3}
Graph y 5 2} Ïx . Then state its domain and range.
–2
x
0.63
y
–1
0
0.5
0
1
O
x
1
2
–0.5
–0.63
The radicand of a cube root can
be any real number, so the
domain is all real numbers .
You can see from the graph that
the range is all real numbers .
y
y 5 221 3 x
1
Checkpoint Graph the function. Then state its domain
and range.
3}
}
2. y 5 22Ïx
1. y 5 2 Ï x
y
y
y 5 23 x
O
x
1
22
1
O
1
x
y 5 22 x
domain and range:
all real numbers
domain: x ≥ 0,
range: y ≤ 0
GRAPHS OF RADICAL FUNCTIONS
}
3}
To graph y 5 aÏ x 2 h 1 k or y 5 a Ï 2 h 1 k:
}
3}
Step 1 Sketch the graph of y 5 aÏ x or y 5 a Ïx .
Step 2 Determine the values of h and k. Translate the
graph h units horizontally and k units
vertically . If h > 0, translate the graph
to the right ; if h < 0, translate it to the left .
If k > 0, translate the graph up ; if k < 0,
translate it down .
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Your Notes
Graph a Square Root Function
Example 3
}
Graph y 5 Ï x 2 1 1 2. State its domain and range.
}
y
1. Sketch the graph of y 5 Ïx .
2. Determine the values of h
and k. Because h 5 1 , you
}
translate the graph of y 5 Ïx
1 unit right . Because
k 5 2 , you translate the
graph 2 units up .
y5 x2112
1
O
x
1
The domain is x ≥ 1 . The range is y ≥ 2 .
Graph a Cube Root Function
Example 4
3}
Graph y 5 Ï x 1 3 2 2. State its domain and range.
3}
1. Sketch the graph of y 5 Ï x .
y
2. Determine the values of h and
k. Because h 5 23 , you
3}
translate the graph of y 5 Ï x
3 units left . Because
k 5 22 , you translate the
graph 2 units down .
1
x
1
y 53x 1 3 2 2
The domain and range are all real numbers .
Checkpoint Graph the function. State domain and range.
}
3}
3. y 5 Ï x 1 3 1 2
4. y 5 Ïx 1 1
y
y
y 5 x 1 3 12
y 5 3x 11
1
O
1
1
x
O
1
x
Homework
domain: x ≥ 23,
range: y ≥ 2
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domain and range:
all real numbers
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