Math Concepts Explained

The first graphing manipulation I will demonstrate is shifting a given graph vertically, and how you can identify this transformation by looking at the equation. This is also called vertical translation.

Let's look at a simple equation first.... y = x. This equation represents a line that goes through (0,0), (1,1), (2,2), etc. Now, what if I asked you to draw the graph represented by the new equation y = x + 2. Some students may suggest to make a table of values and then plot the points on the graph... never a bad way to do things, but not necessarily the easiest.

Let's think about things SLIGHTLY differently. In the first equation, let's think about y specifically as being the height (and not just a different variable), and so the height for a given x value will always be equal to that x value. Not a huge change in our thinking, but it may help some to see graphical changes easier. Now, let's apply this thinking to the second equation, and we see that the height is always going to be 2 greater than the x value.

Now, let's spice things up a bit. Let's look at a function (which, recall, is still the same as saying "y").

f(x) = x^2

You may recognize this equations as being the most basic equation that describes a parabola that opens up. The lowest point on the graph is (0,0).


Now, if I said to draw the graph of f(x) = x^2 + 1, apply the same type of logic, but keep in mind one VERY IMPORTANT THING. The change in height is dictated by the single number that is not associated with the x variable... this may be clearer after another example, but let us focus on this one for the moment. You should hopefully be able to see that this change in our graph will result in a shift up of 1 unit.


One final example, hopefully to clear things up for good. Let's take a complicated function:

f(x) = (x-4)^3 and f(x) = (x-4)^3 + 5


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. So in the end, the second equation above looks exactly like this one, only shifted 5 units up:

Remember to combine terms if necessary, so that you are left with a single number to add to the x term (and whatever operator is acting on it). You will quickly find that vertical translations of graphs are far simpler than they may sound at first!