Graph Transformations and Daylight Saving Time
by David Radcliffe
Saturday was the last day of Daylight Saving Time in the United States. On this day, most of set our clocks back an hour, and we enjoyed an extra hour of sleep. I say “most of us” because Arizona, Puerto Rico, Hawaii, U.S. Virgin Islands and American Samoa do not observe Daylight Saving Time.
Some people find Daylight Saving Time to be confusing, or at least hard to remember. The phrase “spring forward, fall back” is meant to remind us to set our clocks ahead one hour in the spring, and set it one hour back in the fall. One possible reason for the confusion is that there are two different ways to think about Daylight Saving Time. The usual way to express it is that we set the clocks ahead in the spring, and set them back in the fall. We might call this the clock’s point of view. The other point of view is that we must wake up an hour earlier in the spring, but we are permitted to sleep an hour later in the fall.
We encounter a similar concept when we transform the graph of a function by shifting it to the left or right. In algebra, we learn that replacing x with x−1 in an equation will shift the graph one unit to the right, and replacing x with x+1 will shift the graph one unit to the left. This is very perplexing. Surely, subtracting 1 should be a move to the left, and adding 1 should be a move to the right. Why is everything backwards when it comes to graph transformations?
But just as with Daylight Saving Time, there are two ways to think about graph transformations. When we replace x with x−1, we usually say that the graph moves one unit to the right. But an alternative interpretation is that the coordinate system moves one unit to the left, and the graph stays still! To put it another way, suppose you start with the graph of y = f(x), but then you subtract one to each of the number labels on the x-axis. After some thought, you will see that the same graph now describes the equation y = f(x−1).
These two ways of viewing a transformation are called “alibi” and “alias”. An alibi transformation moves the points (“alibi” is Latin for “in another place”.) An alias transformation does not move the points, but only renames them. The difference between an alias and an alibi is illustrated by the hilarious train scene gag in the movie Top Secret (1984).
Sleeping an hour later is an alibi transformation — we are shifting our “sleep curves” one hour to the right. Setting the clock an hour back is an alias transformation — the time coordinate is shifted to the left.