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.

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

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.

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