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What Are Quadratic Functions?

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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.

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