Range of quadratic functions

In this article, we will learn how to find the range of quadratic functions.

In other words, we will learn how to determine the set of all possible outputs of a given quadratic function.

We want to find the range of the function $f(x)=-2(x+3)^2+7$f, left parenthesis, x, right parenthesis, equals, minus, 2, left parenthesis, x, plus, 3, right parenthesis, squared, plus, 7.

Finding the range of a function, just by looking at its formula, is pretty difficult! Actually, it's not even that easy to tell whether a single specific value is a possible output!

For instance, is $y=9$y, equals, 9 a possible output of $f$f?

In order to answer that question, we need to substitute $f$f's formula into $f(x)=9$f, left parenthesis, x, right parenthesis, equals, 9 and solve. If we find a solution, then $y=9$y, equals, 9 is a possible output. Otherwise, it isn't.

However, it's not possible to perform this check for every possible output, because they are infinite! This article will show two possible solution methods to work around this problem.

It turns out graphs are really useful in studying the range of a function. Fortunately, we are pretty skilled at graphing quadratic functions.

Here is the graph of $y=f(x)$y, equals, f, left parenthesis, x, right parenthesis.

Now it's clearly visible that $y=9$y, equals, 9 is not a possible output, since the graph never intersects the line $y=9$y, equals, 9.

Let's perform similar checks for a couple more $y$y-values.

So we saw how we can check whether a given value is a possible output using a graph. A graph can actually tell us the entire range of possible outputs!

For instance, the graph of $y=f(x)$y, equals, f, left parenthesis, x, right parenthesis shows that $7$7 (the $y$y-coordinate of the vertex) is the maximum $y$y-value that the function outputs. Furthermore, since the parabola opens down, every $y$y-value below $7$7 is also a possible output.

In other words, the range of $f$f is all $y$y-values less than or equal to $7$7. This is it! Mathematically, we can write the range of $f$f as ${\{} y\in \mathbb{R} ~|~ y\leq 7 {\}}$left brace, y, \in, R, space, vertical bar, space, y, is less than or equal to, 7, right brace.

Consider the function $g(x)=(x-4)^2-5$g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 4, right parenthesis, squared, minus, 5 which is graphed below.

At this point, you may ask yourselves, "Do we always have to draw the graph when we want to find the range?" and you will be right in doing so! Laziness is a great motivation for finding better ways to solve problems.

Let's think about the work we did above and look for a pattern.

It turns out all we need to know in order to determine the range of a quadratic function is the $y$y-value of the vertex of its graph, and whether it opens up or down.

This is easy to tell from a quadratic function's vertex form, $y = \purpleC{a}(x-h)^2 + \blueD{k}$y, equals, start color #aa87ff, a, end color #aa87ff, left parenthesis, x, minus, h, right parenthesis, squared, plus, start color #11accd, k, end color #11accd. In this form, the vertex is at $y=\blueD{k}$y, equals, start color #11accd, k, end color #11accd, and the parabola opens $\purpleC{\text{up}}$start color #aa87ff, start text, u, p, end text, end color #aa87ff when $\purple{a}>0$start color #9d38bd, a, end color #9d38bd, is greater than, 0 and $\purpleC{\text{down}}$start color #aa87ff, start text, d, o, w, n, end text, end color #aa87ff when $\purpleC{a}<0$start color #aa87ff, a, end color #aa87ff, is less than, 0.