### Maximum and Minimum Point

- 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.
- The minimum/maximum point can be determined from the equation in this form $f(x)=a(x+p{)}^{2}+q$ .

**Minimum Point**

- The quadratic function f(x) has a minimum value if
**a**is positive. - The quadratic function f(x) has a minimum value when (x + p) = 0
- The minimum value is equal to q.
- Hence the minimum point is (-p, q)

**MaximumPoint**

- The quadratic function f(x) has a maximum value if a is negative.
- The quadratic function f(x) has a maximum value when (x + p) = 0
- The maximum value is equal to q.
- 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)

$\begin{array}{l}f(x)={\left(x-3\right)}^{2}+7\\ a=1,p=-3,q=7\\ \\ a>0,\text{thequadraticfunctionhasaminimumpoint}\\ \\ \text{Minimumpoint}\\ =(-p,q)\\ =(3,7)\end{array}$

(b)

$\begin{array}{l}f(x)=-5-3{\left(x+15\right)}^{2}\\ a=-3,\text{}p=15,\text{}q=-5\\ \\ a0,\text{thequadraticfunctionhasamaximumpoint}\\ \\ \text{Maximumpoint}\\ =(-p,q)\\ =(-15,-5)\end{array}$