Form 4 - Quadratic Function

The Graph of a Quadratic Function
The general form of a quadratic function is y = ax^2+bx+c. We have already
studied the graph of the parabola y = x^2. 

The graph of a quadratic function is called a parabola. The vertex is the
point where the parabola achieves its minimum, if it is upward facing, or its
maximum, if it is downward facing.

General form: ax^2+bx+c=0

where a, b, c are constant and a≠0

There are two cases in this topic:

-          a>0(graph is min)

-          a<0(graph is max)

 

for futher notes please click this link :http://www.4shared.com/office/Djks38I9/Quadratic_Functionsnew.html


exersice :  http://www.scribd.com/doc/78990749/quadratic-function-question

Form 4 -Quadratic Equations

In this subtopic you will learn to :
1. Understand the concept of quadratic equation and its roots
1.1 Recognise a quadratic equation and express it in general form. 









Understand the concept of quadratic equation and its roots
The general form of a quadratic equation is ax2 + b x + c = o. where a ,b, and c
are constants and a ≠ 0  . The highest power of the unknown, ( x ) , is 2.

example:

2x + 4 = 0 is not a quadratic equation because the highest power of x is 1

x^2 + 4 = 0 is a quadratic equation because the highest power of x is 2

x^3 + 2x = 4 is not a quadratic equation because the highest power of x is 3

Recognise a quadratic equation and express it in general form

x^2 – 2x = 3 ( is not a general form)
x^2 – 2x – 3 = 0 (is in general form)
Conclusion is, you should know how to rearrange the quadratic equation in general form  ax^2+bx+c=0

2. Understand the concept of quadratic equations
2.1 Determine the roots of a quadratic equation by
        a) factorisation;
        b) completing the square
         c) using the formula.
2.2 Form a quadratic equation from given roots


Determine the roots of a quadratic equation by:

factorisation	completing the square	Using a formula
ax2 + b x + c = o can be factorised completely by converting on the left hand side as a product of two linear factor. (x-a)(x-b) = 0 x-a=0 or x-b=0 x=a or x=b	completing the square is Converting an expression or equation into the "perfect square" form. Converting from general form to perfect square form i.e. y = ax2 + bx + c to y = (x + 1/2a) 2 - (1/2 a)2+ c Eg.  x2 - 8x+5 = (x-4)2 -16+ 3 =(x-4)2-13 It used to find the roots of quadratic equations when the quadratic equations , cannot factorise.	Quadratic Equations can also be solved by using the formula as follows :   where a ,b, c are constants and related to ax2 + bx + c =0

Form a quadratic equation from given roots

1. If a and b are the roots of a quadratic equation then

    x = a or x = b

    x – a = 0 or x – b = 0

   (x – a)(x – b) = 0, hence x ² – (a + b)x + ab = 0

   Therefore , the quadratic equation with roots P and q is

     x² – ( a+b) x + ab = 0

2. The Step of forming a quadratic equation from given roots are

i. Find the sum of the roots

ii. Find the product of the roots

iii. Form a quadratic equation by writing in a following form

        x² – ( sum of the roots ) x + product of the roots = 0

3. Understand and use the conditions for quadratic equations to have
     a) two different roots;
     b) two equal roots;
     c) no roots / no real roots
3.1 Determine types of roots of quadratic equations from the value of b2 − 4ac.


Determine types of roots of quadratic equations from the value of b^2 − 4ac.
For the quadratic equation ax^2 + b x + c = o, the discriminant of the equation is b^2 − 4ac
Types of roots of quadratic equations from the value of b^2 - 4ac
(i) b^2 4ac > 0 ….Two different roots ( the roots are distinct)
(ii) b^2 4ac = 0 …Two same roots
(iii) b^2  4ac < 0 …. No real roots

exercise:  http://www.4shared.com/office/L-9hIAry/QUADRATIC_EQUATION.html