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 - Quadratic Equations - Completing the Square

Final method of solving quadratic equations is Completing the Square. By completing the square for :
 ax² + bx + c, you will generate the Quadratic Formula by solving for x.

The general method consists of changing the quadratic in the form ax² + bx + c into (x+d)²+ e =0

where d = b/2 and e = b² / 4 
* This method requires a = 1, so if the coefficient of x is not 1 . Divide by a to get it to be 1.
* Then rewrite the Formula in the form shown above. Solve for x by :
1) Taking e to the otherside and squarerooting it, and then subtracting b.

My image notes show two examples. 1) How to derive the quadratic formula 2) A numerical example :

Completing the Square Notes

     

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.




 

C1- Algebra and Functions - Quadratic Functions

Now we learn about Quadratic Functions. They're arguably one of the most important types of functions, and you will need to be able to solve them and master the concept, as they will pop up later in other modules. A Quadratic equation, have a degree of 2, takes the form :

ax^2 + bx + c = 0 , where a,b,c are real numbers.

We can solve, quadratic equations using three methods : a) Factorisation, b) Completing the Square and c)The Quadratic Formula.

Quadratic Functions when sketched are of a parabola shape ( a U shape), the coefficient of x^2 will decide the shape of the parabola. If a is negative, the shape will be an upside down U, if positive it will be a U.

The solutions of a quadratic equations are called ROOTS. If you're dealing with a quadratic inequality we call the roots (CRITICAL VALUES).

1) Factorisation... Let ax^2 + bx + c = 0

find two numbers that multiply to give c , and add to give b. (These numbers must be the same for the product and sum)
*Make sure you take care, when you have negative numbers. (Remember a negative * negative = positive)

If you find two numbers that satisfy the product for c, and sum of b: simply put it in the form :

(x + ...) (x + ...) = 0  (... are the two numbers, it does not matter which order they are put)

To find the solution simply put (x +...) = 0, for both of them and solve for x. I shall provide an examples on my
handwritten image.

C1- Algebra and Functions - Rationalising the Denominator

Now that we can manipulate surds, by adding/subtracting/expanding and simplifying. Our next concept, is Rationalising the Denominator. This simply means, making the denominator of a fraction "surd free". There are 3 general rules outlined in the image below with the some examples :

Rationalise Denominator Notes

C1 - Algebra and Functions - Surds

Our next topic is in C1 is Surds. Surds are simply numbers left in the the root form, they are irrational numbers. Firstly we need to know a few rules of surds.

I shall just post images of my scanned written notes, as they are clearer and concise. With Examples.

C1 - Algebra and Functions - Laws of Indices

In this First Post for C1. I shall be posting some notes and examples about the 1st topic in Algebra and Functions. This is about the laws of indices and surds.

Laws of Indices.
A variable raised to the power, is called an indice / index. We have certain rules we can apply, to simplify / manipulate, usually if the base of the power is the same.

1) x^a * x ^b = x^(a+b)   (X to the power of a multiplied by X to the power of b) = X to the power of (a+b)

2)x^a /  x^b = x^(a-b)   (X to the power of a divided by X to the power of b) = X to the power of (a-b)

3) (x^a)^b = x^(a*b) (X^a to the power of b) = X to the power of (a*b)

*** For the 3rd rule, the b will be adjacent to the a, I can't write it in the handwriting form. If there is a number in front of the x, this number will be raised to the power of b.

e.g. (3x^4)*2 = This means 3x to the power of 4 , the 2 is outside the bracket.

The answer would be 3 to the power of 2,multiplied by, x to the power of (4*2) =  (9x^8)

4) (x/m)^n = x^n / m^n (x divided by m, all the power of n = x to the power of n divided by m to the power of n).

General Facts

Anything to the power of 0 will be 1. I mean anything from a number to a variable.

Anythign to the power of 1 will be itself. e.g. 2^1 = 2 ,   (x^2) to the power of 1 = x^2

Other Rules
I have uploaded an image file explaining the rules, as it is not appropriate to write them up, as the typos can be hard to understand.

Handwritten Notes on Laws of Indices (Printable)

If any problem understanding the notes, or feedback email : mathematicsthesolver@gmail.com

C1 Introduction

C1. The first module of AS, of any Alevel Maths Course. Suppose to be the easiest, it generally forms on from the A-A* topics of gcse. However, you don't have a calculator... so brush up on general arithmetic skills, and fractions. Topics :

1)  Algebra and Functions
*Laws of Indices, Surds, Rationalising the Denominator, Quadratic Equations (Discriminant, Completing the Square, Quadratic Formula,Factorisation), Sketching Graphs and Transformations of Graphs.

2) Coordinate Geometry
*Equation of a Straight Line, Gradients, Perpendicular Lines

3) Sequences and Series
*Terms, nth Term, Arithmetic Series, Sum of Terms and Stigma Notations.

4)Differentiation
*Differentiating basic functions, and relation to Gradients.

5)Integration
*Indefinite Integration

Introduction to A-Level Maths

Hello and Welcome to my blog for A-Level Maths. This blog shall cover material / notes from C1 - C4, updating weekly. I am focusing for now on Edexcel's new specification (from 2008 onwards), and hopefully later on will go on to OCR and AQA.

Firstly, I'd like to make clear, if hardwork is put it at any stage, the good results will be the output. It does not really matter whether you have an A* or a C at gcse. Alevel does require independant study, and Practice makes Maths, not rewriting notes hundreds of times. I stress, that it is important to understand a key concept, rather than trying to solve and generalise examples from textbooks. Here is the specification link for Edexcel :

http://www.edexcel.com/migrationdocuments/GCE%20New%20GCE/UA024850%20GCE%20in%20Mathematics%20issue%202%20180510.pdf

Here is the link to the Formula Booklet, you will have access to in the exams :

http://www.edexcel.com/migrationdocuments/GCE%20New%20GCE/N38210A-GCE-Mathematical-Formulae-Statistical-Tables.pdf

For A-Level Maths (Edxcel), you will have to take 4 core modules {C1,C2,C3,C4}. And two applied modules from {D1,D2,M1,M2,S1,S2}. Usually candidates choose D,M,S 1 then D,M,S 2. But you are allowed to take part 1 of two modules, if your school permits.

*C2,C3,C4 are synoptic modules, each requiring candidates to understand content from previous modules.

 AS - (C1,C2 ,1st applied module)
A2 - (C3,C4, 2nd applied module)

I will write up model solutions on request by email. So email me your problems on mathematicsthesolver@gmail.com  {ONLY C1- C4 Problems}.