We shall consider two forms of the quadratic functions.
A  General form.
Quadratic functions in their general form are written as
f(x) = a x^{ 2} + b x + c
where a, b and c are real numbers such that a not equal to zero.
The first
derivative of f is given by
f '(x) = 2 a x + b
Let us analyze the sign of f' and hence determine any maximum or minimum point and the intervals of increase and decrease. f '(x) is positive if
2 a x + b > 0
add b to both sides of the inequality to obtain
2 a x > b
We now need to consider two cases and continue solving the inequality above.
case 1: a > 0
We divide both sides of the inequality by 2 a and obtain
x >  b / 2a
We now use a table to analyze the sign of f ' and whether f is increasing over a given interval.
The quadratic function with a > 0 has a minimum point at (b/2a , f(b/2a)) and the function is decreasing on the interval (infinity , b / 2a) and increasing over the interval (b / 2a , + infinity).
case 2: a < 0
We divide both sides of the inequality by 2 a but because a is less than 0, we need to change the symbol of inequality
x <  b / 2a
We now analyze the sign of f ' using the table below
The quadratic function with a < 0 has a maximum point at (b/2a , f(b/2a)) and the function is increasing on the interval (infinity , b / 2a) and decreasing over the interval (b / 2a , + infinity).
B  Vertex form.
Quadratic functions in their vertex form are written as
f(x) = a (x  h)^{ 2} + k
where a, h and k are real numbers with a not equal to zero.
The first derivative of f is given by
f '(x) = 2 a (x  h)
We analyze the sign of f' using a table. f '(x) is positive if
a (x  h) > 0
We need to consider two cases again and continue solving the inequality above.
case 1: a > 0
We divide both sides of the inequality by a and solve the inequality
x > h
The table below is used to analyze the sign of f '.
The quadratic function with a > 0 has a minimum at the point (h , k) and it is decreasing on the interval (infinity , h) and increasing over the interval (h , + infinity).
case 2: a < 0
We divide both sides of the inequality by a but we need to change the symbol of inequality because a is less than 0.
x < h
We analyze the sign of f ' using the table below
The quadratic function with a < 0 has a maximum point at (h , k) and the function is increasing on the interval (infinity , h) and decreasing over the interval (h , + infinity).
Example 1: Find the extremum (minimum or maximum) of the quadratic function f given by
f(x) = 2 x^{ 2}  8 x + 1
Solution to Example 1.

We first find the derivative
f '(x) = 4 x  8
f '(x) changes sign at x = 8 / 4 = 2. The leading coefficient a is positive hence f has a minimum at (2 , f(2)) = (2 , 7) and f is decreasing on (infinity , 2) and increasing on (2 , + infinity). See graph below to confirm the result obtained by calculations.
Example 2: Find the extremum (minimum or maximum) of the quadratic function f given by
f(x) =  (x + 3)^{ 2} + 1
Solution to Example 2.

The derivative is given by
f '(x) =  2 (x + 3)
f '(x) changes sign at x = 3 . The leading coefficient a is negative hence f has a maximum at (3 , 1) and f is increasing on (infinity , 3) and decreasing on (3 , + infinity). See graph below of f below.
Exercises on Properties of Quadratic Functions.
For each quadratic function below find the extremum (minimum or maximum), the interval of increase and the interval of decrease.
a) f(x) = x^{ 2} + 6 x
b) f(x) = x^{ 2}  2 x + 3
c) f(x) = x^{ 2}  5
d) f(x) = (x  4)^{ 2} + 2
e) f(x) = x^{ 2}
Answers to Above Exercises.
a) minimum at (3 , 9)
decreasing on (infinity , 3)
increasing on (3 , + infinity)
b) maximum at (1 , 4)
increasing on (infinity , 1)
decreasing on (1 , + infinity)
c) minimum at (0 , 5)
decreasing on (infinity , 0)
increasing on (0 , + infinity)
d) maximum at (4 , 2)
increasing on (infinity , 4)
decreasing on (4 , + infinity)
e) maximum at (0 , 0)
increasing on (infinity , 0)
decreasing on (0 , + infinity)
More on applications of differentiation
applications of differentiation
