How To Find The Range Of Parabolas

In mathematics, some quadratic functions create what's known as a parabola when you graph them. Though the width, location and direction of the parabola will vary based on the specific function being graphed, all parabolas are generally "U" shaped (sometimes with a few extra fluctuations in the middle) and are symmetric on both sides of their center point (also known as the vertex.) If the function you're graphing is an even-ordered function, you're going to have a parabola of some type.

When working with a parabola, there are a few details that are useful to calculate. One of these is the domain of a parabola, which indicates all possible values of ​x​ included at some point along the parabola's arms. This is a pretty easy calculation because the arms of a true parabola continue spreading out forever; the domain includes all real numbers. Another useful calculation is the parabola range, which is a little trickier but not that difficult to find.

The domain and range of a parabola essentially refer to which values of ​x​ and which values of ​y​ are included within the parabola (assuming that the parabola is graphed on a standard two-dimensional ​x​-​y​ axis.) When you draw a parabola on a graph, it might seem weird that the domain includes all real numbers because your parabola most likely looks like just a little "U" there on your axis. There's more to the parabola than you see, however; each arm of the parabola should end with an arrow, indicating that it continues on to ∞ (or to −∞ if your parabola faces down.) This means that even though you can't see it, the parabola will eventually spread out in both directions large enough to encompass every possible value of ​x​.

The same doesn't hold true on the ​y​ axis, however. Look at your graphed parabola again. Even if it's placed at the very bottom of your graph and opens upward to encompass everything above it, there are still lower values of y that you simply haven't drawn on your graph. In fact, there's an infinite number of them. You can't say that the parabola range includes all real numbers because no matter how many numbers your range includes, there are still an infinite number of values that fall outside of the range of your parabola.

A range is a representation of values between two points. When you're calculating the range of a parabola, you only know one of those points to start with. Your parabola will go on forever either up or down, so the end value of your range is always going to be ∞ (or −∞ if your parabola faces down.) This is good to know, because it means that half of the work of finding the range is already done for you before you even start calculating.

If your parabola range ends at ∞, where does it start? Look back at your graph. What is the lowest value of ​y​ that is still included in your parabola? If the parabola opens down, flip the question: What is the highest value of ​y​ that is included in the parabola? Whatever that value is, there's the beginning of your parabola. If, for example, your parabola's lowest point is on the origin – the point (0,0) on your graph – then the lowest point would be ​y​ = 0 and the range of your parabola would be ​[0, ∞)​. When writing range, use brackets [ ] for numbers included in the range (such as the 0) and parentheses ( ) for numbers that aren't included (such as ∞, since it can never be reached).

What if you just have a formula, though? Finding the range is still pretty easy. Convert your formula to the standard polynomial form, which you can represent as

\(y = ax^n + ... + b\)

for these purposes, use a simple equation such as

\(y = 2x^2 + 4\)

If your equation is more complex than this, simplify it to the point that you have any number of ​x​s to any number of powers with a single constant (in this example, 4) at the end. This constant is all that you need to discover the range because it represents how many spaces up or down the y axis your parabola shifts. In this example it would move up 4 spaces, whereas it would move down four if you had

\(y = 2x^2 – 4\)

Using the original example, you can then calculate the range to be [4, ∞), making sure to use brackets and parentheses appropriately.