The 2 methods are:

- graphical
- mathematical re-expression of equation

I shall deal with both methods here.

Example:

**y = x**(a quadratic equation)

^{2}- 2x + 3**1)**

**Graphical method**

By plotting the curve generated by the equation, we are able to visually identify the lowest point and get the co-ordinates directly.

The demerit of this method is having to plot the graph out and the co-ordinate values are as accurate as plotting accuracy.

The graph is drawn as shown in diagram 1.

Diagram 1 Equation plotted to identify the lowest point on the curve

From diagram 1, you are able to see that the lowest point is at x = 1 and y = 2. (The point marked with the red cross). This graphical method is straight forward but needs effort in sketching the curve accurately.

**2) Mathematical re-expression method**

This method involves more manipulation of the equation rather than sketching graph.

It makes use of the "

**Completing the Square**" method in factorisation to extract out the lowest point. For a review of the "Completing the Square" method, please refer to this

**.**

__link__y = x

^{2}- 2x + 3 can be re-written as

y = (x

^{2}- 2x + c

^{2}) -c

^{2}+ 3

**Note: The "-c**

^{2}" term outside the closing bracket is to retain the originality of the equation.The purpose is to form a y expression with an order of 2 or (x -c)

^{2}for reason which will be clear later.

What is this "c" to be ?

By following the principle of the "Completing the Square" technique, "c" can be seen to be (2/2).

Therefore the new equation becomes

y = [x

^{2}- 2x + (2/2)

^{2}] - (2/2)

^{2}+ 3

Now, here comes the

**important concept of this method**to locate the lowest point.

From the new y expression, you see that to make the y value the lowest possible, the only thing you can do is to make the x value equal to the "c" value or the (2/2) value.

That is, [x - (2/2)] = 0 ==> x = (2/2) for lowest value of y.

When x = (2/2) = 1, y must then be -(2/2) + 3 = +2.

Therefore, the answer to the lowest co-ordinates is x = 1 and y = 2.

This is the same as for the graphical method. However, it does not involve tedious plotting, beside more accurate numerical values is ensured.

.

## No comments:

Post a Comment