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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) = ax2 + bx + c,

where a, b and c are constants. The constants a and b are called coefficients. The term ax2 is called a second-degree term or quadratic term, the term bx 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) : The graph is smiling, and the function has a minimum (vertex).

  • When a < 0 (negative) : The graph is sad, and the function has a maximum (vertex).

Example 1

You have the quadratic function

f(x) = 0.5x2˘x˘3

Determine if the parabola has a maximum or a minimum.

You see that the coefficient in front of x2 is a positive number (a > 0). Therefore, the graph smiles and have a minimum.

Example of a minimum of quadratic function

Example of a minimum of quadratic function

Example 2

You have the quadratic function

f(x) = x2˘x + 12

Determine if the parabola has a maximum or a minimum.

In this case, the coefficient in front of x2 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

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) = ax2 + bx + 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 = b 2a

y-value of the maximum or minimum point:

y = f ( b 2a)

Example 3

Determine if the graph of f(x) = x2 4x + 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:

x = (4) 2 1 = 4 2 = 2 y = f(2) = 22 (4 2) + 4 = 4 8 + 4 = 0

x = (4) 2 1 = 4 2 = 2 y = f(2) = 22 4 2 + 4 = 4 8 + 4 = 0

The minimum is then (2, 0).

Upon inspection of f(x) = x2 4x + 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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What Are Polynomial Functions?