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2.10.1 Maximum and Minimum Value of Quadratic Functions

Maximum and Minimum Point

  1. A quadratic functions f(x)=ax2+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)

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+7a=1,p=−3,q=7a>0, the quadratic function has a minimum pointMinimum point=(−p,q)=(3,7)

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

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