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Let us discuss how to identify the graphs of a quadratic function.

*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 quadratic function \(f: \mathbb{R} \rightarrow \mathbb{R}\) with domain \(x \in \mathbb{R}\) and range \(f(x) \in [0, \infty)\) is defined by \(f(x) = x^2\).

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 a domain \(x \in \mathbb{R}\) and a range \(f(x) \in (- \infty, 0]\).

The graphical representation of the quadratic function is given by:

Important!

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