### Theory:

A function is of the form $$f(x) = ax^2 + bx + c$$, where $$a$$, $$b$$ and $$c$$ are constants, and $$a \neq 0$$ is called a quadratic function.
Example: The trajectory of cables in the bridge results in a parabolic path. A parabola represents a quadratic function.
A parabola is symmetric with respect to a line called the axis of symmetry. The point of intersection of the parabola and the axis of symmetry is called the vertex.
Important!
For a quadratic equation, the axis is given by $$x = \frac{-b}{2a}$$ and the vertex is given by $$\left(\frac{-b}{2a}, \frac{-\Delta}{4a} \right)$$ where $$\Delta = b^2 - 4ac$$ is the discriminant of the quadratic equation $$ax^2 + bx + c = 0$$.
A parabola usually forms a "U" shaped curve. Depending on the value of $$a$$ in the general equation of the parabola $$y = ax^2$$, the parabolas open upwards or downwards and vary in width.

Let us consider the simple graph $$y = x^2$$. Let us compare the above graph with graphs having the higher value of $$a$$. Here, the graph $$y = x^2$$ is broader than the graph $$y = 8x^2$$.

Similarly, compare the graph $$y =x^2$$ with graphs having lower values of $$a$$. Here, the graph $$y = x^2$$ is narrower than the graph $$y = \frac{1}{6}x^2$$.

Important!
1. The greater the value of $$a$$, the narrower is the parabola.

2. The lesser the value of $$a$$, the wider is the parabola.
Reference:
https://pixabay.com/photos/golden-gate-bridge-san-francisco-388917/