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.

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