## Horizontal Transformation: Sliding a Function Sideways

**Question**

This is the graph of :

Sketch the graph of .

**Solution**

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.