Horizontal Transformation: Sliding a Function Sideways
This is the graph of :
Sketch the graph of .
The change from to is called a horizontal transformation; it slides the function sideways. The graph of lies one unit to the right of the graph of . This may be counterintuitive, so let’s look at some numbers. At each x-value, the graph of takes the y-value that the graph of 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, graphs one unit to the left of . Note that this is different from which graphs one unit up from because the 1 that gets added to the function means 1 gets added to every y-value. Similarly, graphs one unit below If you don’t believe me on any of these, try playing around with values of a simple function to see what happens.