## HOW TO FIND THE MINIMUM OR MAXIMUM VALUE OF A FUNCTION IN VERTEX FORM

How to Find the Minimum or Maximum Value of a Function in Vertex Form :

Here we are going to see,  how to find the maximum and minimum value of a function in vertex form.

## Complete the Square and Find the Minimum or Maximum Value of the Quadratic Function

To find the vertex form of the parabola, we use the concept completing the square method.

Vertex form of a quadratic function :

y = a(x - h)2 + k

In order to find the maximum or minimum value of quadratic function, we have to convert the given quadratic equation in the above form.

Minimum value of parabola :

If the parabola is open upward, then it will have minimum value

If a > 0, then minimum value of f is f(h)  =  k

Maximum value of parabola :

If the parabola is open downward, then it will have maximum value.

If a < 0, then maximum value of f is f(h)  =  k

## How to Find the Minimum or Maximum Value of a Function in Vertex Form - Examples

Question 1 :

For the given function f(x) = x2 + 7x + 12

(a) Write f(x) in the form k(x + t)2 + r.

(b) Find the value of x where f(x) attains its minimum value or its maximum value.

(c) Find the vertex of the graph of f.

Solution :

Let y  =  x2 + 7x + 12

y  =  x2 + 2x⋅(7/2) + (7/2)2 - (7/2)2 + 12

y  =  (x + (7/2))2 + 12

By comparing it with vertex form, we get the value of k . Since it is positive, the parabola is open upward. So it will minimum value.

(b)  It has minimum value when x = -7/2

(c)  Vertex of the parabola is (-7/2, 12)

Question 2 :

For the given function f(x) = 5x2 + 2x + 1

(a) Write f(x) in the form k(x + t)2 + r.

(b) Find the value of x where f(x) attains its minimum value or its maximum value.

(c) Find the vertex of the graph of f.

Solution :

Let y  =  5x2 + 2x + 1

y  =  5(x2 + x) + 1

y  =  5 [x2 + 2 ⋅ ⋅ (1/2) + (1/2)2 - (1/2)2] + 1

y  =  5 [x + (1/2)]2 - (1/4)] + 1

y  =  5 [x + (1/2)]2 - (5/4) + 1

y  =  5 [x + (1/2)]2 - (1/4)

By comparing it with vertex form, we get the value of k . Since it is positive, the parabola is open upward. So it will minimum value.

(b)  It has minimum value when x = -1/2

(c)  Vertex of the parabola is (-1/2, -1/4)

Question 3 :

For the given function f(x) = −2x2 + 5x − 2

(a) Write f(x) in the form k(x + t)2 + r.

(b) Find the value of x where f(x) attains its minimum value or its maximum value.

(c) Find the vertex of the graph of f.

Solution :

Let y  = −2x2 + 5x − 2

y  =   −2[x2 - (5/2)x] − 2

y  =   −2[x2 - (5/2)x] − 2

y  =  -2 [x2 - 2 ⋅ ⋅ (5/4) + (5/4)2 - (5/4)2] - 2

y  =  -2 [x - (5/4)]2 - (25/16)] - 2

y  =  -2 [x - (5/4)]2 + (25/8) - 2

y  =  -2 [x - (5/4)]2 + (9/8)

By comparing it with vertex form, we get the value of k . Since it is negative, the parabola is open downward. So it will maximum value.

(b)  It has maximum value when x = 5/4

(c)  Vertex of the parabola is (5/4, 9/8)

After having gone through the stuff given above, we hope that the students would have understood "How to Find the Minimum or Maximum Value of a Function in Vertex Form".

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