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3.2 Maximum and Minimum Value of Quadratic Functions

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Maximum and Minimum Point

  1. A quadratic functions f ( x ) = a x 2 + b x + 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)

Maximum Point

  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 ) = ( x − 3 ) 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 ) = − 5 − 3 ( x + 15 ) 2 a = − 3 ,   p = 15 ,   q = − 5 a < 0 ,  the quadratic function has a maximum point Maximum point = ( − p , q ) = ( − 15 , − 5 )