Quadratic functions

Quadratic functions are functions that can be written in the standard form f(x) =
ax2 + bx + c, where a ≠ 0 and a, b, and c are all constants.

The standard form has 3 different types of terms:

  • ax2 is called the quadratic term
  • bx is called the linear term
  • c is called the constant term.

Notice that the condition that a ≠ 0 ensure that every function has a quadratic term, but not necessarily a linear term or a constant term. If a = 0, the function has no quadratic term. In that case, the function is not a quadratic function.

Graph of quadratic functions

Here is how the graph of quadratic functions will look like. Notice the axis of symmetry and the vertex that can either be a maximum or a minimum.

Examples of quadratic functions

f(x) = 2x2 + 3x + -4

f(x) = 2x2 - 3x + 5

f(x) = -x2 + x + 100

f(x) = 3x2 + 6x

f(x) = 5x2 + -4

f(x) = 6x2 

f(x) = (x + 2)(x+3)

f(x) = 2(x - 3)2 + 4

Notice that f(x) = (x + 2)(x+3) = x2 + 3x + 2x + 6 = x2 + 5x + 6 

Notice also that f(x) = 2(x - 3)2 + 4 = 2(x2 - 6x + 9) + 4 = 2x2 - 12x + 18 + 4 =

2x2 - 12x + 22

A function may appear to be a quadratic function when in fact it is not quadratic. 

For example, f(x) = 4(x2 + x)  -  4(x2 + 8) is not a quadratic function.

f(x) = 4(x2 + x)  -  4(x2 + 8)

f(x) = 4x2 + 4x  -  4x2 + -32

f(x) = 4x + -32

As you can see, this function is linear.

The three forms of quadratic functions

General form

f(x) = ax2 + bx + c, where a, b, and c are real numbers, is called the general form of a quadratic function.

Factored form

f(x) = (ax + b)(cx + d), where a, b, c, and d are real numbers, is called the factored form of a quadratic function.

Vertex form

f(x) = a(x - h)2 + k, where a, h, and k are real numbers, is called the vertex form of a quadratic function.

Recent Articles

  1. Rational Numbers - Definition and Examples

    Mar 15, 23 07:45 AM

    Rational numbers
    To learn about rational numbers, write their decimal expansion, and recognize rational numbers that are repeating decimals and terminating decimals.

    Read More

  2. Area of a Trapezoid - Definition, Formula, and Examples

    Mar 13, 23 07:52 AM

    Area of a trapezoid1
    Learn how to get the area of a trapezoid using a rectangle and a triangle, the formula, and also when the height of the trapezoid is missing.

    Read More

Tough algebra word problems

100 Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Math quizzes

 Recommended

Math vocabulary quizzes