Shifting graphs horizontally (also known as horizontal translation) is slightly different from vertical translation, but still pretty straight-forward. Perhaps it would be helpful to review my posting on vertical shifts of graphs. Recall from that section: “Picture all the complex stuff that is happening to x as being one chunk of the height component, and then when you add the + 5 to the equation, you are really just adding an additional height chunk to the total height for a given x.” I think this simplification condenses the rest of that post down quite nicely.
Shifting Graphs – Horizontal Translation
Now, to shift a graph horizontally, you include the shift amount with x. So, whatever action was being done just to x before, now you do that same thing to x plus the shift amount. Make sense? Probably not.
Check out the example below that hopefully demonstrates this better than I can explain with words.
If you want to shift the original function of f(x) = x2 + 4 by 3 units, it becomes f(x) = (x-3)2 + 4.
Can you see what I mean by including the shift amount WITH x. The ‘square’ function acts on the entire (x-3) term. This will cause the graph to shift 3 units to the RIGHT. This may seem somewhat counter-intuitive, but it is correct. Subtracting terms from x shift the graph to the right, whereas adding terms to x will translate them to the left.
In this example, x-3 causes a horizontal translation of the graph 3 units right… if it were x+3, it would translate the graph 3 units left. Here is a bit of a trick you can use to help you recall the direction of the shift caused by the signs. It may be easier to remember this by analyzing the “x and shift amount”, letting this small term equal to 0, and then solving for x. The result will show you how many units to move, and in what direction. Like this:
x – 3 = 0
x = 3 (shift 3 units right)
x + 3 = 0
x = (-3) (shift 3 units left)
That shows you how far over, and in what direction, the new x values are! Technically, this is a way of finding a zero of the graph, but that is another post for another day. For now, I think it’s a helpful trick to apply at this stage!
I hope these postings on graph manipulations are helpful. Horizontal translation of functions and their graphs is still quite simple, albeit with the trick with the signs that you don’t have to worry about with vertical translation.