**Summary**

- A quadratic equation is in the form ,
- A quadratic function has a minimum value when a > 0
- A quadratic function has a maximum value when a < 0
- Vertex is a point on the where the curve is either maximum or minimum
- To find the values of the maximum or minimum point we use the method called ”Completing the square method”.
- is the vertex, we get from the form obtained:

after applying completing the square method.

An expression which is written in the form where *a*, *b* and *c* are constants and , is called a ”Quadratic Function”.

We know that a graph for the equation , when a > 0, is a parabola with a minimum value as shown in Fig 1.

However, when a < 0, the graph of the function has a maximum value as shown in Fig 2.

Here *P* and *Q* are called the vertex of the parabola.

Therefore, we now know that the maximum or minimum value of a quadratic function depends on the coefficient of in the function.

#### Completing the Square Method

To find the values of the maximum or minimum point we use the method called ”Completing the square method”.

Let’s look at a generic example to be able to understand how this method works.

Suppose we have a quadratic function where *a*, *b* and *c* are constants.

Firstly, we make 1 the coefficient of :

We now add and subtract a term :

The first three terms can be simplified to become :

Taking the LCM of the last two terms we get:

Multiplying *a* outside the brackets with the terms inside, we get:

From the expression obtained above, we know that a square quantity is always positive, therefore:

i) If a > 0, then the function has a minimum value

ii) If a < 0, then the function has a maximum value

iii) The graph of the function above is always symmetrical in shape about the line

iv) The point is called vertex of the parabola.

#### Graph of a quadratic function

To be able to sketch the graph of a quadratic function, we must remember the following four points:

1. Check the value of coefficient of to know the shape of the curve, whether it has a minimum point (i.e Fig 1) or a maximum point (i.e Fig 2).

2. Find the vertex (point where the curve is maximum or minimum) of the curve

3. Substitute y = 0 in the equation of a function find *x* intercept.

4. Substitute x = 0 in the equation of a function find *y* intercept.

#### Example #1

Q. Find the vertex of the curve

*Solution:*

To begin with we can see that since the coefficient of is < 0, this curve will have a maximum value and the shape of the curve will be like in Fig 2.

We know a = -3, b = -2 and c = -4

To find the vertex use the formula:

** Ans**

#### Example #2

Q. Solve by completing the square method

*Solution:*

Since already has a coefficient 1, so we now add and subtract

a = 1, b = 6 and c = 5

Equate the expression to 0 to find the values of *x:*

This gives us two values of *x*:

** Ans**

and

** Ans**