Posts

Horizontal Transformation: Sliding a Function Sideways

Horizontal Transformation: Sliding a Function Sideways

Question

This is the graph of y=f\left ( x \right )

Sketch the graph of y=f(x-1).

Solution

The change from y=f\left ( x \right ) to y=f(x-1) is called a horizontal transformation; it slides the function sideways. The graph of y=f(x-1) lies one unit to the right of the graph of y=f\left ( x \right ). This may be counterintuitive, so let’s look at some numbers. At each x-value, the graph of y=f(x-1) takes the y-value that the graph of y=f\left ( x \right ) takes one unit earlier. Some values on the two graphs look like this:

The y-values from the original graph have all chugged along to correspond to higher x-values than they did before.

At any x-value on the graph, the function  operates on the  value 1 unit to the left of that x-value. That has the effect of moving the curve 1 unit to the right.

Similarly, y=f(x+1) graphs one unit to the left of y=f\left ( x \right ). Note that this is different from y=f\left ( x \right )+1, which graphs one unit up from y=f\left ( x \right ) because the 1 that gets added to the function means 1 gets added to every y-value. Similarly, y=f\left ( x \right )-1 graphs one unit below f\left ( x \right ). If you don’t believe me on any of these, try playing around with values of a simple function to see what happens.