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>

C1 - Algebra and Functions - Quadratic Equations (Solving using Formula)

One of the methods, apart from Factorisation is using the Quadratic Formula. Usually, when a quadratic equations cannot be factorised, it is a good idea to use the Formula. Though the formula works for all quadratic equations. We have to use the form ax^2 + bx + c. Identity our coefficients and simply substitute into the formula which is :

There's nothing else to know I reckon for the Quadratic Equation. Though I will show you how to derive it by Completing the Square (which i will cover in the next post). Another fundamental point that comes out of the Formula, is the b^2 - 4ac, this is called the Discriminant. We shall look it later, it is important because we can determine how a quadratic function may look, and the number and types of roots assosciated with the equation.

Quadratic Formula Notes

TO SEE ANY OF MY IMAGE NOTES IN FULL, right click and click on view image.