Algebra
Textbooks
Boundless Algebra
Quadratic Functions and Factoring
Introduction to Quadratic Functions
Algebra Textbooks Boundless Algebra Quadratic Functions and Factoring Introduction to Quadratic Functions
Algebra Textbooks Boundless Algebra Quadratic Functions and Factoring
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 9
Created by Boundless

What is a Quadratic Function?

Quadratic equations are second order polynomials, and have the form $f(x)=ax^2+bx+c$.

Learning Objective

  • Describe the criteria for, and properties of, quadratic functions


Key Points

    • A quadratic function is of the form $f(x)=ax^2+bx+c$, where a is a nonzero constant, b and c are constants of any value, and x is the independent variable. 
    • The solutions to a quadratic equation are known as its zeros, or roots.

Terms

  • quadratic function

    A function of degree two.

  • dependent variable

    Affected by a change in input, i.e. it changes depending on the value of the input.

  • independent variable

    The input of a function that can be freely varied.

  • vertex

    The minimum or maximum point of a quadratic function. 


Full Text

The single defining feature of quadratic functions is that they are of the second order, or of degree two. This means that in all quadratic functions, the highest exponent of $x$ in a non-zero term is equal to two.  A quadratic function is of the general form:

$f(x)=ax^2+bx+c$

where $a$, $b$, and $c$ are constants and $x$ is the independent variable.  The constants $b$and $c$ can take any finite value, and $a$ can take any finite value other than $0$.  

A quadratic equation is a specific case of a quadratic function, with the function set equal to zero:

$ax^2+bx+c=0$

When all constants are known, a quadratic equation can be solved as to find a solution of $x$.  Such solutions are known as zeros.  There are several ways of finding $x$, but these methods will be discussed later.

Differences Between Quadratics and Linear Functions

Quadratic equations are different than linear functions in a few key ways. 

  • Linear functions either always decrease (if they have negative slope) or always increase (if they have positive slope). All quadratic functions both increase and decrease.
  • With a linear function, each input has an individual, unique output (assuming the output is not a constant).  With a quadratic function, pairs of unique independent variables will produce the same dependent variable, with only one exception (the vertex) for a given quadratic function.
  • The slope of a quadratic function, unlike the slope of a linear function, is constantly changing.

Forms of Quadratic Functions

Quadratic functions can be expressed in many different forms. The form written above is called standard form. Additionally

$f(x)=a(x-x_1)(x-x_2)$

is known as factored form, where $x_1$ and $x_2$ are the zeros, or roots, of the equation. These are $x$ values at which the function crosses the y-axis (and thus where $y$ equals zero). 

The vertex form is displayed as:

$f(x)=a(x-h)^2+k$

where $h$ and $k$ are respectively the coordinates of the vertex, the point at which the function reaches either its maximum (if $a$ is negative) or minimum (if $a$ is positive).

[ edit ]
Edit this content
Prev Concept
Fitting a Curve
The Quadratic Formula
Next Concept
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.