Operations on Functions

Math
  • Study Guide
Summary

Translations

Summary 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, y1) is a point on the graph of f (x), (x, y2) is a point on the graph of f (x) + 2, and (x, y3) is a point on the graph of f (x) - 2, then y2 = y1 + 2 and y3 = y1 - 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 (x1, y) is a point on the graph of f (x), (x2, y) is a point on the graph of f (x + 2), and (x3, y) is a point on the graph of f (x - 2), then x2 = x1 - 2 and x3 = x1 + 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) = x2 + 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) f1(x) = f (x) + 4 = x2 + 2x + 4
b) f2(x) = f (x) - 4 = x2 + 2x - 4
c) f3(x) = f (x + 4) = (x + 4)2 +2(x + 4) = x2 +8x + 16 + 2x + 8 = x2 + 10x + 24
d) f4(x) = f (x - 4) = (x - 4)2 +2(x - 4) = x2 -8x + 16 + 2x - 8 = x2 - 6x + 8