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.

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).

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:

Solutions:

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.