X Co-ordinate Changes: Translations

At this point, we know how to manipulate a function's y values through the use of constant. These changed its y values or its vertical appearance. Now will change the graph's horizontal shape and position. These are given the general form of y =  f(x+c), where f(x) is the original function, and c is a constant value.

Let's take a look at two sample functions  to see how it works.
 y = f (x - 5) 1) Generate a Table of Values for y = (x - 5)2 The values shown in yellow are used for the original graph. The values shown in blue are used for the new graph. 2) Plot the Graph of the Original Function and the New Function 3) Compare the Graphs The original graph is shown in blue, and the new graph is shown in green. How does it compare? The green parabola has the same amount of data points as the blue one. However, we are using much higher x values than before to get the same y values. Its shape looks exactly the same, but it looks like its been moved to the right. y = f(x - 9) 1) Generate a Table of Values for y = (x + 9)2 The values shown in yellow are used for the original graph. The values shown in blue are used for the new graph. 2) Plot the Graph of the Original Function and the New Function 3) Compare the Graphs The original graph is shown in blue, and the new graph is shown in red. How does it compare? Notice that the x-values in the table of values has shifted. The values we are using are a lot less than in the original function. It appears as though the new graph is shited to the left.The red function looks exactly like the original blue graph.

The overall effect appears to be that adding a constant term to the x value of a function changes its position. If the constant is positive, the graph moves to the left; if the constant is negative, the graph moves to the right. Result: y = f(x+c) is similar to y = f(x) but is moved left or right depending on the value of c.

Why does this happen?

When we evaluate the new function, f(x+c), we are changing what the function acts on. We are changing the actual arguments (inputs) of the function.

Here are some questions to ask:

1) If I know the value of f(0) for the original graph, where does it appear in the new graph for f(x+c)? What new x value do I need?

That same value of f(0) in the original graph should appear at x = - c. When we substitute x = - c into the new function we should get f (x+c) = f ( [-c] + c) = f (0).

2) If I know the value of f(3) for the original graph, where does it appear in the new graph for f(x+c)? What new x value do I need?

The same value of f(3) in the original graph should appear at x = 3 - c. When we substitute x = 3 - c into the new function we should get f (x+c) = f ( [3 - c] + c) = f (3).

2) If I know the value of f(-1) for the original graph, where does it appear in the new graph for f(x+c)? What new x value do I need?

The same value of f(-1) in the original graph should appear at x = -1 - c. When we substitute x = -1 - c into the new function we should get f (x+c) = f ( [-1 - c] + c) = f (-1).

We should see the same values of f(x) but moved around to different x values

Notice that in each of the answers, the value -c appeared. If c is positive, we need to move each point on the graph c units to the left (towards -c). This provides a kind of correction factor. Since adding a positive number will increase the value of the argument, we can move it towards a lower value to negate its effects. If c is negative, move the graph towards -(-c) or +c  to get the same y values. Since adding a negative number decreases the value of the argument, we can move it towards a greater value to counter its effects.This causes a shift to the right (in the positive x direction). Adding a constant factor c to the argument of a function x causes this shift.