# How to Find Domain and Range of a Quadratic Function

The domain of a quadratic function in standard form is always all real numbers, meaning you can substitute any real number for *x*. The range of a function is the set of all real values of *y* that you can get by plugging real numbers into *x*.

## Domain and Range of Quadratic Functions

Hi, and welcome to this video about the domain and range of quadratic functions! In this video, we will explore: How the structure of quadratic functions defines their domains and ranges and how to determine the domain and range of a quadratic function

Let’s get started!

Before we begin, let’s quickly revisit the terms domain and range.

The domain of a function is the set of all possible inputs, while the range of a function is the set of all possible outputs.

The structure of a function determines its domain and range. Some functions, such as linear functions (for example fx=2x+1), have domains and ranges of all real numbers because any number can be input and a unique output can always be produced. On the other hand, functions with restrictions such as fractions or square roots may have limited domains and ranges (for example \(fx=\frac{1}{2x}\). x cannot be 0 because the denominator of a fraction cannot be 0).

Let’s see how the structure of quadratic functions defines and helps us determine their domains and ranges.

Here’s the graph of fx = x^{2}. Quadratic functions together can be called a family, and this particular function the parent, because this is the most basic quadratic function (i.e. not transformed in any way). We can use this function to begin generalizing domains and ranges of quadratic functions.

To determine the domain and range of any function on a graph, the general idea is to assume that they are both real numbers, then look for places where no values exist.

Let’s talk about domain first. Since domain is about inputs, we are only concerned with what the graph looks like horizontally. To see the domain, let’s move from left-to-right along the x-axis looking for places where the graph doesn’t exist.

As you can see, there are no places where the graph doesn’t exist horizontally. The domain of this function is all real numbers. *In fact, the domain of all quadratic functions is all real numbers*!

Now for the range. We’ll use a similar approach, but now we are only concerned with what the graph looks like vertically.

As you can see, outputs only exist for y-values that are greater than or equal to 0. In other words, there are no outputs below the x-axis. We would say the range is all real numbers greater than or equal to 0.

Let’s generalize our findings with a few more graphs.

The range for this graph is all real numbers greater than or equal to 2

The range here is all real numbers less than or equal to 5

The range for this one is all real numbers less than or equal to -2

And the range for this graph is all real numbers greater than or equal to -3

As you can see, the turning point, or vertex, is part of what determines the range. The other is the direction the parabola opens. If a quadratic function opens up, then the range is all real numbers greater than or equal to the y-coordinate of the range. If a quadratic function opens down, then the range is all real numbers less than or equal to the y-coordinate of the range.

Graphs can be helpful, but we often need algebra to determine the range of quadratic functions. Sometimes, we are only given an equation and other times the graph is not precise enough to be able to accurately read the range.

So, let’s look at finding the domain and range algebraically. There are three main forms of quadratic equations. Our goals here are to determine which way the function opens and find the y-coordinate of the vertex.

When quadratic equations are in standard form, they generally look like this: fx = ax^{2} + bx + c. If a is positive, the function opens up; if it’s negative, the function opens down. In this form, the y-coordinate of the vertex is found by evaluating \(f(\frac{-b}{2a})\). For example, consider this function:

\( fx=-2x^2+8x-3\),

\(\frac{-b}{2a}=\frac{-8}{2(-2)}=\frac{-8}{-4}=2\),

\(f(2)=-8+16-3=5\).

a is negative, so the range is all real numbers less than or equal to 5.

When quadratic equations are in vertex form, they generally look like this: \(fx=a(x-h)^2+k\). As with standard form, if a is positive, the function opens up; if it’s negative, the function opens down. The vertex is given by the coordinates (h,k), so all we need to consider is the k. For example, consider the function \(fx=3(x+4)^2-6\). *a* is positive and the vertex is at -4,-6 so the range is all real numbers greater than or equal to -6.

Sometimes quadratic functions are defined using factored form as a way to easily identify their roots. For example: \(fx=a(x-b)(x-c)\). As with the other forms, if a is positive, the function opens up; if it’s negative, the function opens down. One way to use this form is to multiply the terms to get an equation in standard form, then apply the first method we saw. We can also apply the fact that quadratic functions are symmetric to find the vertex. We know the roots, and therefore, the locations of the x-intercepts. Horizontally, the vertex is halfway between them. Once we know the location of the vertex – the x-coordinate – all we need to do is substitute into the function to find the y-coordinate. For example, consider the function \(fx=-2(x+4)(x-2)\). The x-intercepts are at -4 and 2 and the vertex is located at \(\frac{-4+2}{2}=-1\) (simply take the “average” of the x-intercepts). We’re going to plug it into our original equation: \(f(-1)=-23-3=18\). Since a is negative, the range of all real numbers is less than or equal to 18.

Ok, let’s do a quick review before we go.

- Domain is the set of input values, while range is the set of output values.
- To determine the domain and range of any function on a graph, the general idea is to assume that they are both real numbers, then look for places where no values exist.
- And finally, when looking at things algebraically, we have three forms of quadratic equations: standard form, vertex form, and factored form.

Thanks for watching, and happy studying!