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# 2.10.1 Maximum and Minimum Value of Quadratic Functions

### Maximum and Minimum Point

1. A quadratic functions $f\left(x\right)=a{x}^{2}+bx+c$ can be expressed in the form $f\left(x\right)=a\left(x+p{\right)}^{2}+q$ by the method of completing the square.
2. The minimum/maximum point can be determined from the equation in this form $f\left(x\right)=a\left(x+p{\right)}^{2}+q$ .
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. $f\left(x\right)=\left(x-3{\right)}^{2}+7$
b. $f\left(x\right)=-5-3\left(x+15{\right)}^{2}$