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Range of quadratic functions

Learn how you can find the range of any quadratic function from its vertex form.
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.

Let's study an example problem

We want to find the range of the function f(x)=2(x+3)2+7.
In this article, just as we're used to referring to inputs of a function with the letter x, we will refer to the outputs of a function with the letter y. For instance, y=7 is the output of f for an input of x=3 (this is just another way of saying f(3)=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 a possible output of f?
In order to answer that question, we need to substitute f's formula into f(x)=9 and solve. If we find a solution, then y=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.

Solution method 1: The graphical approach

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).
Now it's clearly visible that y=9 is not a possible output, since the graph never intersects the line y=9.
Let's perform similar checks for a couple more y-values.
Question 1Question 2
Is y=5 a possible output of f?
Choose 1 answer:

Is y=50 a possible output of f?
Choose 1 answer:

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) shows that 7 (the y-coordinate of the vertex) is the maximum y-value that the function outputs. Furthermore, since the parabola opens down, every y-value below 7 is also a possible output.
In other words, the range of f is all y-values less than or equal to 7. This is it! Mathematically, we can write the range of f as {yR | y7}.

Your turn!

Consider the function g(x)=(x4)25 which is graphed below.
What is the range of g ?
Choose 1 answer:

Solution method 2: The algebraic approach

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.
Our first function, f(x)=2(x+3)2+7, had a parabola that opened down and whose vertex was at y=7. In consequence, its range was all y-values less than or equal to 7.
Our second function, g(x)=(x4)25, had a parabola that opened up and whose vertex was at y=5. In consequence, its range was all y-values greater than or equal to 5.
It turns out all we need to know in order to determine the range of a quadratic function is the 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=a(xh)2+k. In this form, the vertex is at y=k, and the parabola opens up when a>0 and down when a<0.

Your turn

Use what you've learned to find the range of h(x)=12(x3)2+2.

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