The graph of a function can be moved up, down, left, or right by adding to or subtracting from the output or the input.

Adding to the output of a function moves the graph up.
Subtracting from the output of a function moves the graph down.
Here are the graphs of
*y* = *f* (*x*)
,
*y* = *f* (*x*) + 2
, and
*y* = *f* (*x*) - 2
. Note
that if
(*x*, *y*
_{1})
is a point on the graph of
*f* (*x*)
,
(*x*, *y*
_{2})
is a point on
the graph of
*f* (*x*) + 2
, and
(*x*, *y*
_{3})
is a point on the graph of
*f* (*x*) - 2
,
then
*y*
_{2} = *y*
_{1} + 2
and
*y*
_{3} = *y*
_{1} - 2
. For example,
(1, 2)
is on the graph
of
*f* (*x*)
,
(1, 4)
is on the graph of
*f* (*x*) + 2
, and
(1, 0)
is on the graph
of
*f* (*x*) - 2
.

Graphs of
*f* (*x*)
,
*f* (*x*) + 2
, and
*f* (*x*) - 2

While adding to the input increases the function in the
*y*
direction, adding to
the input decreases the function in the
*x*
direction. This is because
the function must compensate for the added input. If the function outputs "7"
when "3" is input, and we input
*x* + 2
, the function will output "7" when
*x* = 1
.

Thus, adding to the input of a function moves the graph left, and
subtracting from the input of a function moves the graph right.
Here are the graphs of
*y* = *f* (*x*)
,
*y* = *f* (*x* + 2)
, and
*y* = *f* (*x* - 2)
. Note
that if
(*x*
_{1}, *y*)
is a point on the graph of
*f* (*x*)
,
(*x*
_{2}, *y*)
is a point on
the graph of
*f* (*x* + 2)
, and
(*x*
_{3}, *y*)
is a point on the graph of
*f* (*x* - 2)
,
then
*x*
_{2} = *x*
_{1} - 2
and
*x*
_{3} = *x*
_{1} + 2
. For example,
(1, - 2)
is on the
graph of
*f* (*x*)
,
(- 1, - 2)
is on the graph of
*f* (*x* + 2)
, and
(3, - 2)
is on
the graph of
*f* (*x* - 2)
.

Graphs of
*f* (*x*)
,
*f* (*x* + 2)
, and
*f* (*x* - 2)

A shift of the graph up, down, left, or right, without changing the shape, size, or dimensions of the graph, is called a translation.

*Examples*: If
*f* (*x*) = *x*
^{2} + 2*x*
, what is the equation if the graph is
shifted:

a) 4 units upSolutions:

b) 4 units down

c) 4 units left

d) 4 units right

a)f_{1}(x) =f(x) + 4 =x^{2}+ 2x+ 4

b)f_{2}(x) =f(x) - 4 =x^{2}+ 2x- 4

c)f_{3}(x) =f(x+ 4) = (x+ 4)^{2}+2(x+ 4) =x^{2}+8x+ 16 + 2x+ 8 =x^{2}+ 10x+ 24

d)f_{4}(x) =f(x- 4) = (x- 4)^{2}+2(x- 4) =x^{2}-8x+ 16 + 2x- 8 =x^{2}- 6x+ 8

We can think of translating a graph as creating a "new origin." When we add or
subtract a constant
*k*
to the output, we move the
origin up and down. When we
add or subtract a constant
*h*
to the input, we move the origin left or right,
because we change the value of
*x*
which yields
*f* (*x* + *h*) = *f* (0)
. We then
graph the function on the new origin.