Y Co-ordinate Changes: Translation

Given any function, we want to be able to change its features so that we can match other similar looking graphs to it. While each of the examples presented on these webpages is a quadratic function (a parabola), these general principles apply to all functions. The function y = x2 was chosen because it is fairly easy to generatel a table of values to check the graphs. You can download a sheet of graph paper from here if you like try some yourself. (Requires GSView to view).

There are two general changes that can be made to a graph's y values.  The first change is to change a graph's vertical position. This 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) + 6 1) Generate a Table of Values for y = x2 + 6 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? It appears that the new graph is six units higher than the original graph. Its shape is still the same. y = f(x) - 2 1) Generate a Table of Values for y = x2 - 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? It appears that the new graph is two units lower than the original graph. Its shape is still the same.

The overall effect appears to be that adding a constant term to a function raises each y-value of that function, but doesn't change its shape. Result: y = f(x) + c is the same shape as y = f(x) but it has been shifted vertically by c units.

Why does this happen?

When we evaluate the new function, we first evaluate f(x) at any value of x. In this case, the value of f(x) remains the same for that x value. Then by adding on the constant term at the end of the equation, we add c to f(x). This means each value of x has the value of f(x) at the point, with an additional c added on to it.

On To Scaling Y-Coordinates
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