6.5 Graphing
Square Root &
Cube Root
Functions
Algebra II
First, let’s look at the parent
graphs.
y3 x
y x
4.5
4
4
16, 4
3
27, 3
3.5
2
3
8, 2
9, 3
1
1, 1
2.5
2
0
4, 2
-30
-20
-10
0, 0
0
-1 -1, -1
1.5
1
1, 1
-8, -2 -2
0.5
-27, -3
0
-3
0, 0
0
2
4
6
8
10
12
14
16
18
-4
10
20
30
Now, what happens when there is a number
in front of the radical?
y2 x
y  33 x
10
12
-27, 9
8
10
25, 10
-8, 6 6
8
4
16, 8
-1, 3
2
6
9, 6
0
-30
4
-20
-10
0, 0
0
10
20
30
-2
4, 4
1, -3
-4
2
1, 2
-6
8, -6
-8
0
0, 0
0
27, -9
5
10
15
20
25
* Notice the graph goes thru
the points (0,0) and (1,2).
30
-10
* Notice the graph goes thru
the points (-1, 3), (0,0), & (1,-3).
Generalization
ya x
Always goes thru
the points (0,0) and
(1,a).
y  a3 x
Always goes thru the
points (-1,-a), (0,0),
and (1,a).
Ex. 1: Graph y  4
x
Goes thru the points (0,0) and (1,a).
Since a=-4, the graph will pass thru
(0,0) and (1,-4)
0
0, 0
0
5
10
15
20
25
30
1, -4
-5
-10
9, -12
-15
16, -16
-20
-25
25, -20
Now, what happens when there are
numbers added or subtracted inside and/or
outside the radical?
y  a x  h  k or y  a 3 x  h  k
Step 1: Find points on the parent graph
ya
x or y  a 3 x
Step 2: Shift these points h units horizontally (use
opposite sign) and k units vertically (use same
sign).
Ex. 2: Describe how to obtain the graph of
y  x  2  1 from the graph of y  3 x
3
Shift all the points from
To the right 2 and up 1.
y3 x
Ex. 3: Graph y  2 x  4  1
(x-value – 4)
(y-value -1)
table for y  2 x
x
0
Now, shift these points to the left 4
and down 1.
y
0
6
x
y
5
5, 5
4
1
2
-4
-1
3
0, 3
2
4
9
4
-3
1
6
0
3
5
5
-3, 1
1
0
-6
-4
-2
-4, -1
0
-1
-2
2
4
6
3
y


2
x 3  2
Ex. 4: Graph
(x-value + 3)
(y-value + 2)
table for y  23 x
Now, shift these points to the right 3
and up 2.
x
y
-27
6
x
y
4
-24
8
-8
10
-24, 8
8
-5,
66
-1
2
-5
6
0
0
2
4
1
-2
3
2
8
-4
4
0
4
2, 4
2
3, 2
0
27
-6
11
-2
30
-4
-30
-20
-10
4, 0
0
-2
-4
-6
10
20
30
40
11, -2
30, -4
Ex. 3 & 4: State the domain and range of
the functions in the last 2 examples.
x-values
y-values
3
y  2 x  4 1
y  2 x  3  2
6
10
5
5, 5
-24, 8
8
4
-5,
66
3
0, 3
4
2
-3, 1
2
1
-20
-10
0
-4
-2
0
-4, -1
3, 2
0
-30
-6
2, 4
2
4
-2
6
4, 0
0
10
20
30
40
11, -2
-1
-4
30, -4
-2
-6
Domain:
Range:
x  4
y  1
The graph has a beginning
point of (-4,-1).
all real numbers
Range: all real numbers
Domain:
The graph doesn’t have a beginning or
ending point.
(Meaning all x & y-values are possible.)
Assignment