# Operations on Functions

### Translations

#### Translations

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 + 2x , what is the equation if the graph is shifted:

a) 4 units up
b) 4 units down
c) 4 units left
d) 4 units right
Solutions:
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.

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