>Types of Functions>What Are Quadratic Functions?

What Are Quadratic Functions?

A polynomial function where the highest exponent is 2 is called a quadratic function. The graph of a quadratic function is called a parabola. It looks either like a happy smile, or a sad frown. You can see different parabolas below.

Theory

Quadratic Function

The quadratic function looks like this:

f (x) = a x^{2} + b x + c,

where $a$ , $b$ and $c$ are constants. The constants $a$ and $b$ are called coefficients. The term $a x^{2}$ is called a second-degree term or quadratic term, the term $b x$ is called a first-degree term or linear term, and $c$ is called the constant term.

Below you see a picture of the graphs of four different quadratic functions. Note that all have either a maximum or minimum.

Graph of four quadratic functions plotted together in one coordinate system

Rule

Parabola (Maximum and Minimum)

When $a > 0 (positive) \Rightarrow ⋃$ : The graph is smiling, and the function has a minimum (vertex).
When $a < 0 (negative) \Rightarrow ⋂$ : The graph is sad, and the function has a maximum (vertex).

Example 1

You have the quadratic function

f (x) = 0.5 x^{2} - x - 3

Determine if the parabola has a maximum or a minimum.

You see that the coefficient in front of $x^{2}$ is a positive number ( $a > 0$ ). Therefore, the graph smiles and have a minimum.

Example of a minimum of quadratic function

Example 2

You have the quadratic function

f (x) = - x^{2} - x + 12

Determine if the parabola has a maximum or a minimum.

In this case, the coefficient in front of $x^{2}$ is a negative number ( $a < 0$ ). Therefore, the graph will face downwards and have a maximum, as in the figure below:

Example of a maximum of quadratic function

There are several methods to use to find the maximum or minimum. You can use the derivative, a sign chart, or the method that follows here:

Rule

Finding the Maximum or Minimum of a Parabola

When you have the quadratic function

f (x) = a x^{2} + b x + c,

you can find the $x$ -values and $y$ -values of the vertex like this:

The $x$ -value of the maximum or minimum point:

x = - \frac{b}{2 a}

$y$ -value of the maximum or minimum point:

y = f (- \frac{b}{2 a})

Example 3

Determine if the graph of $f (x) = x^{2} - 4 x + 4$ has a maximum or a minimum, and find this point

The function $f (x)$ has $a = 1$ , $b = - 4$ and $c = 4$ . Since $a = 1 > 0$ in the expression, you know that the graph is smiling and you have a minimum.

First, find the $x$ -value, and then the $y$ -value, of this minimum:

\begin{array}{l} x & = \frac{- (- 4)}{2 \cdot 1} = \frac{4}{2} = 2 \\ y & = f (2) \\ = 2^{2} - (4 \cdot 2) + 4 \\ = 4 - 8 + 4 \\ = 0 \end{array}

\begin{array}{l} x & = \frac{- (- 4)}{2 \cdot 1} = \frac{4}{2} = 2 \\ y & = f (2) = 2^{2} - 4 \cdot 2 + 4 = 4 - 8 + 4 = 0 \end{array}

The minimum is then

(2, 0)

. Upon inspection of

f (x) = x^{2} - 4 x + 4

, you can see that this is a square and therefore has only one zero. In this case, the zero and the minimum are the same point.

Want to know more?Sign UpIt's free!

What Are Polynomial Functions?

Go back