HOW TO FIND MAXIMUM AND MINIMUM VALUE OF A QUADRATIC EQUATION

The graphical form of a quadratic function will be a parabola (u shpae). In which the maximum and minimum value will be there at vertex.

To find the maximum or minimum value from the quadratic equation, we have the following ways.

(i) Converting into the vertex form

(ii) Using formula

Find the maximum or minimum value of the following function.

Problem 1 :

y = –x² + 2x + 3

Solution :

Method 1 :

y = –x² + 2x + 3

Factoring negative sign.

y = –[x² - 2x - 3]

Write the coefficient of x as a multiple of 2.

y = –[x² - 2 ⋅ x ⋅ 1 + 1² - 1² - 3]

y = –[(x-1)² - 1 - 3]

y = –[(x-1)² - 4]

y = –(x-1)² + 4

It is exactly in the form of 

y = a(x - h)² + k

Here a = -1 < 0, then the parabola opens down.

Vertex is at (1, 4). So, minimum is at x = 1 and the minimum value = 4.

Method 2 :

y = –x² + 2x + 3

a = -1, b = 2 and c = 3

x = -b/2a

x = -2/2(-1)

x = 1

By applying the value of x in the given equation, we get

y = -(1)² + 2(1) + 3

y = -1 + 2 + 3

y = 4

So, the minimum point is at (1, 4).

Minimum value is y = 4

Problem 2 :

y = 2x² + 4x – 3

Solution :

y = 2x² + 4x – 3

Factoring 2, we get

y = 2(x + 1)² - 5

Here (h, k) is (-1, -5).

Vertex is at (-1, -5).

So,

minimum is at x = -1 and 

minimum value = -5.

Problem 3 :

y = –3x² + 4x 

Solution :

y = –3[x² + 4x/3]

Here (h, k) is (2/3, 4/3).

Vertex is at (2/3, 4/3).

So,

minimum is at x = 2/3 and 

minimum value = 4/3