Transformations of
Functions
Graphs of Common Functions
See Table 1.4, pg 184.
Characteristics of Functions:
1. Domain
2. Range
3. Intervals where its increasing, decreasing, or
constant
4. symmetry
Vertical Shifts
Let f be a function and c a positive real number.
•The graph of y = f (x) + c is the graph of y = f (x) shifted c units
vertically upward.
•The graph of y = f (x) – c is the graph of y = f (x) shifted c units
vertically downward.
y = f (x) + c
y = f (x)
c
c
y = f (x)
y = f (x) - c
Example
• Use the graph of y = x2 to obtain the graph of y = x2 + 4.
10
8
6
4
2
-10 -8 -6 -4 -2
2
-2
-4
-6
-8
-10
4
6
8 10
Example cont.
• Use the graph of y = x2 to obtain the graph of y = x2 +
4.
Step 1 Graph f (x) = x2. The graph of the standard quadratic
function is shown.
Step 2 Graph g(x) = x2+4. Because we add 4 to each value of x2
in the range, we shift the graph10of f vertically 4 units up.
8
6
4
2
-10 -8 -6 -4 -2
2
-2
-4
-6
-8
-10
4
6
8 10
Horizontal Shifts
Let f be a function and c a positive real number.
•The graph of y = f (x + c) is the graph of y = f (x) shifted to the left c
units.
•The graph of y = f (x - c) is the graph of y = f (x) shifted to the right c
units.
y = f (x)
y = f (x + c)
c
y = f (x)
c
y = f (x - c)
Text Example
Use the graph of f (x) to obtain the graph of h(x) = (x + 1)2 – 3.
Solution
5
x2.
Step 1 Graph f (x) =
The graph of the
standard quadratic function is shown.
4
3
2
1
-5 -4 -3 -2
1)2.
-1
-1
-2
1
Step 2 Graph g(x) = (x +
Because we
-3
-4
add 1 to each value of x in the
-5
domain, we shift the graph of f
horizontally one unit to the left.
Step 3 Graph h(x) = (x + 1)2 – 3. Because we subtract 3, we shift the
graph vertically down 3 units.
2
3 4
5
Reflection about the x-Axis
• The graph of y = - f (x) is the graph of y = f (x) reflected
about the x-axis.
Reflection about the y-Axis
• The graph of y = f (-x) is the graph of y = f (x)
reflected about the y-axis.
Stretching and Shrinking
Graphs
Let f be a function and c a positive real number.
•If c > 1, the graph of y = c f (x) is the graph of y = f (x) vertically
stretched by multiplying each of its y-coordinates by c.
•If 0 < c < 1, the graph of y = c f (x) is the graph of y = f (x) vertically
shrunk by multiplying each of its y-coordinates by c.
g(x) = 2x2
f (x) = x2
10
9
8
7
6
5
4
3
2
1
-4 -3 -2 -1
h(x) =1/2x2
1
2
3
4
Sequence of Transformations
A function involving more than one
transformation can be graphed by performing
transformations in the following order.
1. Horizontal shifting
2. Vertical stretching or shrinking
3. Reflecting
4. Vertical shifting
Example
• Use the graph of f(x) = x3 to graph g(x) =
(x+3)3 - 4
Solution:
10
8
6
Step 1: Because x is
replaced with x+3, the
graph is shifted 3 units to
the left.
4
2
-10 -8 -6 -4 -2
2
-2
-4
-6
-8
-10
4
6
8 10
Example
• Use the graph of f(x) = x3 to graph g(x) =
(x+3)3 - 4
Solution:
10
8
6
Step 2: Because the
equation is not multiplied
by a constant, no
stretching or shrinking is
involved.
4
2
-10 -8 -6 -4 -2
2
-2
-4
-6
-8
-10
4
6
8 10
Example
• Use the graph of f(x) = x3 to graph g(x) =
(x+3)3 - 4
Solution:
10
8
6
Step 3: Because x remains
as x, no reflecting is
involved.
4
2
-10 -8 -6 -4 -2
2
-2
-4
-6
-8
-10
4
6
8 10
Example
• Use the graph of f(x) = x3 to graph g(x) =
(x+3)3 - 4
10
Solution:
8
6
Step 4: Because 4 is
subtracted, shift the graph
down 4 units.
4
2
-10 -8 -6 -4 -2
2
-2
-4
-6
-8
-10
4
6
8 10