Maximum and Minimum Value of Quadratic Functions

Maximum and Minimum Point

  1. A quadratic functions \(f(x) = a{x^2} + bx + c\) can be expressed in the form \(f(x) = a{(x + p)^2} + q\) by the method of completing the square.
  2. The minimum/maximum point can be determined from the equation in this form (\(f(x) = a{(x + p)^2} + q\)).
Minimum Point
  1. The quadratic function f(x) has a minimum value if a is positive
  2. The quadratic function f(x) has a minimum value when (x + p) = 0
  3. The minimum value is equal to q.
  4. Hence the minimum point is (-p,q)

MaximumPoint
  1. The quadratic function f(x) has a maximum value if a is negative.
  2. The quadratic function f(x) has a maximum value when (x + p) = 0
  3. The maximum value is equal to q.
  4. Hence the maximum point is (-p,q)
Example
Find the maximum or minimum point of the following quadratic equations
a. \(f(x) = {(x - 3)^2} + 7\)
b. \(f(x) = - 5 - 3{(x + 15)^2}\)
Answer:
a.
f(x)= ( x3 ) 2 +7 a=1,p=3,q=7 a>0, the quadratic function has a minimum point Minimum point =(p,q) =(3,7) b.
f(x)=53 ( x+15 ) 2 a=3, p=15, q=5 a<0, the quadratic function has a maximum point Maximum point =(p,q) =(15,5)