Differentiation is used to analyze the properties such as intervals of increase, decrease, local maximum, local minimum of quadratic functions.

We shall consider two forms of the quadratic functions. ## A - Quadratic Function in General form
Quadratic functions in their general form are written as
f(x) = a x^{ 2} + b x + cwhere 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: coefficient a > 0We divide both sides of the inequality by 2 a and obtainx > - 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: coefficient a < 0We divide both sides of the inequality by 2 a but because a is less than 0, we need to change the symbol of inequalityx < - 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 - Quadratic Function in Vertex formQuadratic functions in their vertex form are written asf(x) = a (x - h)^{ 2} + kwhere 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: coefficient a > 0We divide both sides of the inequality by a and solve the inequalityx > 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: coefficient a < 0We 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 1Find the extremum (minimum or maximum) of the quadratic function f given by^{ 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 2Find the extremum (minimum or maximum) of the quadratic function f given by^{ 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 FunctionsFor 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 Exercisesa) 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 differentiationapplications of differentiation |