Quadratic Functions - Definition & Examples

A quadratic function is simply a function which has degree $2$. Check out the explanations for Standard Form and Degree if you need a refresher on this terminology.

Now, when written in standard form, quadratic functions are of the form $$f(x) = ax^2 + bx+c$$ where $a, b,$ and $c$ are constants and $a\neq 0.$

Suppose we're given the function $f(x) = 3x^2-2+5x-x^2+5$. We can write this in standard form by combining terms with the same degree:

$$\begin{align*}3x^2-2+5x-x^2+5 &= (3x^2-x^2) + (-2 + 5) + 5x\\ &= 2x^2+3+5x\end{align*}$$

We want to put it in standard form. We need to have the degrees in decreasing order:

$$\begin{align*}2x^2+3+5x = 2x^2+5x+3\end{align*}$$

So, we know $f(x) = 2x^2+5x+3$.

We can use the standard form to determine many properties of the function. We'll cover this in a bit.

Sometimes we'll need to find the roots of a quadratic function. A root (or zero) of a function is a number, $x$, such that $f(x) = 0$. You may have seen this with completing the square of the quadratic formula. Graphically, this is the same as finding the $x$-intercepts for the function.

This is a graph of $f(x) = x^2-x-2$ in red. The blue dots represent the $x$ intercepts at $x=2$ and $x=-1$. Both of these $x$ values are also roots of $f(x)$ as they both make $f(x)=0$.

We know that the shape of a quadratic function is a parabola, where the coordinates of the vertex is the minimum or maximum. The real roots, real numbers, are the points where the parabola crosses the x-axis.

It's important to distinguish a function from an equation. A function is something of the form $f(x) = ax^2 + bx + c$, which defines the formula in terms of $x$. An equation is when we set the formula equal to something. So $ax^2+bx+c = 0$ is a quadratic equation, not a function.

(Video) An Introduction to Quadratic Functions

by Randy Anderson

Randy Anderson has a good video introduction to quadratic functions.

$\large{\text{Summary}}$

Quadratic functions are quadratic equations that are set up to be graphed. They take the following form: $$f(x)=ax^2+bx+c$$ where $a, b,$ and $c$ are constants and $a\neq 0.$

This is a little bit different than a quadratic equation. Equations are set equal to some constant that you then solve. Functions are defined in terms of a variable, and primarily used for graphing.

Each coefficient in a quadratic function gives us information, but the most important is the $a$ value. If $a > 0$, the parabola opens upwards. If $a < 0$, the parabola faces downwards.

Every parabola has a vertex and an axis of symmetry. The vertex is the point where the graph "turns around". The axis of symmetry is a vertical line that splits the parabola in half.

All of those values can be determined with information given by the function.

Quadratic Functions

Quadratic functions have the standard form: $$f(x)=ax^2+bx+c$$ where $a, b,$ and $c$ are constants and $a\neq 0.$

Quadratic functions have a degree of two (i.e., the highest exponent of the variable is $2$) and when you graph them, they form parabolas. In the following sections, you will learn how to graph quadratic functions and solve quadratic equations!