Theory:

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 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:
 
Q1.png
 
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:
 
Q2.png
 
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
shutterstock_1410089585.jpg