Parent Graphs and Transformations
A picture of a graph can easily be created by using transformations on the parent graph of a
function. This is helpful in saving time by not having to do needless calculations. The following
concept can be applied.
Shift of a Graph:
When the function of the graph is y = ±f(x – h) + k, then a shift of the original graph of y = f(x)
can be created by moving h-units in the x-direction and k-units in the y-direction. If there is an
negative sign in front of the function, then the graph is reflected over the x-axis.
EX.1 f(x) = |x – 3| + 5
10
y = |x-3| + 5
8
6
4
2
y = |x|
5 units up
0
x
3 units right
-2
-4
-6
-8
-10
-5
-4
-3
-2
-1
0
y
1
2
3
4
5
EX.2 f(x) = -(x + 4)2 – 3
15
10
5
f(x) = x2
0
x
3 units down
4 units left
-5
-10
f(x) = -(x+4)2 - 3
-15
-10
The Math Center
-8
■
-6
-4
Valle Verde
-2
0
y
■
2
4
6
Tutorial Support Services
8
■
10
EPCC
Constant Function (y = c)
Linear Function ( y = x )
y
5
25
4
4
20
3
3
15
2
2
10
1
1
5
5
0
x
-1
-1
-5
-2
-2
-10
-3
-3
-15
-4
-4
-20
-5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-4
-3
-2
-1
0
y
1
2
3
4
-25
-5
5
-4
-3
Square Root Function ( y = x1/2 )
Absolute Value Function ( y = |x| )
-2
-1
0
1
2
3
4
5
Cube Function ( y = x3 )
5
5
4
4
100
3
3
50
2
2
1
1
x
0
x
0
x
0
-1
-1
-50
-2
-2
-3
-3
-100
-4
-4
-5
-5
-5
-5
x
0
x
0
Square Function
(y = x2)
y
-4
-3
-2
-1
0
y
1
2
3
4
5
-5
-5
-4
-3
-2
-1
0
y
1
2
3
4
-5
5
-4
-3
150
5
-1
0
y
1
2
3
4
5
Cube Root Function (y = x1/3)
Exponential Function (y = ax)
Logarithmic Function (y = log x)
-2
5
4
4
100
3
3
2
2
50
1
1
0
x
0
x
0
x
-1
-1
-50
-2
-2
-3
The Math Center
-100
■
-3
Valle Verde
■
Tutorial Support Services
■
-4
-4
-5
-5
EPCC
-4
-3
-2
-1
0
y
1
2
3
4
5
-150
-5
-4
-3
-2
-1
0
y
1
2
3
4
5
-5
-5
-4
-3
-2
-1
0
y
1
2
3
4
5