Question

This is the graph of $y=f\left&space;(&space;x&space;\right&space;)$

Sketch the graph of $y=f(x-1)$.

Solution

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