- 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.
Q. Find the vertex of the curve
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:
Q. Solve by completing the square method
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: