- By adding a constant to a function , what is the relation between the graphs of and ?
- By performing a “change of variable” , what is the relation between the graphs of and ?
- How to use this new understanding to gain a deeper understanding of graphs of quadratic functions (i.e., parabolas)?
Introduction
Let’s consider the two quadratic functions and , defined for all real values of . We know what their graphs look like:
The graphs are very similar, down to the horizontal “width”. In fact, drawing them together, we may see that they only differ by a horizontal translation:
Algebraically, one can see that this happens because This hints at the following general fact: doing horizontal shifts to the graph of a function amounts to replacing with “” inside . In this unit, we’ll understand in more detail how to work with this, and also how to deal with vertical shifts, as opposed to horizontal shifts. Since vertical shifts are much easier to understand, that’s where we’ll begin.
Shifting a function vertically
Let’s consider a very simple situation, where we have two functions and . Graphing them, in order, we have that
Clearly, and are directly related via , and seeing their graph together, we have that:
In other words, the graph of was obtained from the graph of by shifting it up exactly by units. This is a very general phenomenon, that happens for any functions who differ by a constant.
- To graph , shift the graph of up units, by adding to the -coordinates of the points on the graph of .
- To graph , shift the graph of down units, by subtracting from the -coordinates of the points on the graph of .
In the above setting, it is useful to call the parent function.
- a.
- .
- b.
- .
- c.
Granted, graphing a linear function poses little to no challenge, but understanding how things work in this setting might offer us some general insight on -intercepts. If , then . Graphing is easier than easy: just a line with slope passing through the origin. And the shift down by units comes last, as you would expect:
Shifting a function horizontally
Consider again the example given in the introduction, where we have and . The first thing we would like to address is a source of frequent confusion when first learning this topic. Namely, we have replaced with in the formula for , but the graph of the modified function ended up shifted to the right, even though one might expect the shift to have happened to the left, due to the negative sign in the factor!
Here is one safe way to think about it: imagine that you are standing on the -axis and, say, at the origin of the cartesian plane, but that the graph of is already drawn. Replacing with does move the -axis to the left. But you, the observer, standing on the -axis, sees the graph move to the right!
Alternatively, compare this with what happened with vertical shifts, but switching the roles of the -axis and -axis. More precisely, start with the graph of , then rotate it by clockwise (this switches the axes). Replacing with now brings the graph down by unit. Finally, rotate everything back by counterclockwise (this undoes the switching of the axes). The resulting graph is obtained from the original one by shifting it to the right, not left.
- To graph , shift the graph of right units, by adding to the -coordinates of the points on the graph of .
- To graph , shift the graph of left units, by subtracting from the -coordinates of the points on the graph of .
As before we’ll continue to call the parent function, whose graph will be drawn in blue, while the graphs of the “child” functions will be indicated in red.
- a.
- .
- b.
- .
Consider this time . Then we have that , which means that to graph , we may take the graph of and shift it to the right by units. We obtain: - c.
- .
As you might be guessing by now, the parent function can be found by just seeking the shifted variable (in this case, ), and replacing it with . Meaning that if , then . Thus, to graph , we can just graph and shift it units to the right. Since we can write , we know that is a parabola which crosses the -axis at and , and it is concave up. Hence:
- Vertical shifts: given the graph of , we can draw the graph of , with , by shifting the graph of up by units. Similarly, the graph of is obtained by shifting the original graph down by units.
- Horizontal shitfs: given the graph of , we can draw the graph of , with , by shifting the graph of by units to the left. Similarly, the graph of is obtained by shifting the original graph by units to the right.