Answer:
Step-by-step explanation:
One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function
g
(
x
)
=
f
(
x
)
+
k
, the function
f
(
x
)
is shifted vertically
k
units.
Graph of f of x equals the cubed root of x shifted upward one unit, the resulting graph passes through the point (0,1) instead of (0,0), (1, 2) instead of (1,1) and (-1, 0) instead of (-1, -1)
Figure 2. Vertical shift by
k
=
1
of the cube root function
f
(
x
)
=
3
√
x
.
To help you visualize the concept of a vertical shift, consider that
y
=
f
(
x
)
. Therefore,
f
(
x
)
+
k
is equivalent to
y
+
k
. Every unit of
y
is replaced by
y
+
k
, so the
y
-
value increases or decreases depending on the value of
k
. The result is a shift upward or downward.
A GENERAL NOTE: VERTICAL SHIFT
Given a function
f
(
x
)
, a new function
g
(
x
)
=
f
(
x
)
+
k
, where
k
is a constant, is a vertical shift of the function
f
(
x
)
. All the output values change by
k
units. If
k
is positive, the graph will shift up. If
k
is negative, the graph will shift down.
EXAMPLE 1: ADDING A CONSTANT TO A FUNCTION
To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. Figure 2 shows the area of open vents
V
(in square feet) throughout the day in hours after midnight,
t
. During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.
Solution
HOW TO: GIVEN A TABULAR FUNCTION, CREATE A NEW ROW TO REPRESENT A VERTICAL SHIFT.
Identify the output row or column.
Determine the magnitude of the shift.
Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.
EXAMPLE 2: SHIFTING A TABULAR FUNCTION VERTICALLY
A function
f
(
x
)
is given below. Create a table for the function
g
(
x
)
=
f
(
x
)
−
3
.
x
2 4 6 8
f
(
x
)
1 3 7 11
Show Solution
Identifying Horizontal Shifts
We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at h
