C1- Algebra and Functions - Quadratic Graphs and The Discriminant

After knowing how to solve the Quadratic Equations by all methods. I shall now introduce some key points of a Quadratic Function, and how it looks on graphs.. while introducing the Discriminant.

The Quadratic Equation has a degree to the power 2 (when we say x squared). We say quadratics are polynomials of the degree 2, in the form of ax^2 + bx + c. Where a,b,c are constants, and a is not 0. The graph of a quadratic is  a parabola, almost a U shape, like this :

The shape of a parabola 

When a (coefficient of  is x²  )is positive, a U shaped parabola will be the graph. When a is negative, it will be an upside down U. This is one of the transformation of functions, I will be showing you in a later post ( which results in a reflection in the x-axis)


















<sup>Where this curve, crosses the x-axis, are known as the roots. These roots are the "solutions" when we solve these functions as equations by equalling them to 0, and using one of the 3 methods.

The Discriminant 
The Discriminant, is an expression, which allows us to find out the nature of the roots of a quadratic equation. It is given by :</sup> 

 <sup>There are 3 possibilities of the discriminant :
* Discriminant > 0 (This means the quadratic has two distinct real roots)
*Discriminant < 0 (This means the quadratic has two distinct complex / no real roots)
*Discriminant = 0 (This means the quadratic has 1 real root / repeared root)

I shall provide all three posibilities with an example, an my image notes, along with some graphs.</sup>