### Theory:

Let us discuss how to identify the graphs of a quadratic function.
A function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ defined by $$f(x) = ax^2 + bx + c$$, $$a \neq 0$$ is called a quadratic function.

Let us further discuss some specific quadratic functions.
Specification 1:
A function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ defined by $$f(x) = x^2$$ is a quadratic function with its domain $$x \in \mathbb{R}$$ and range $$f(x) \in [0, \infty)$$.
The graphical representation of this quadratic function is given by:

Specification II:
A function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ defined by $$f(x) = - x^2$$ is a quadratic function with its domain $$x \in \mathbb{R}$$ and range $$f(x) \in (- \infty, 0]$$.
The graphical representation of quadratic function is given by:

Important!
• The quadratic functions are not one-one functions.
• The equation of motion of the particle travelling under the influence of gravity is an example of the quadratic function of time