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

Quadratic functions in their general form are written as

\[f(x) = ax^2 + bx + c\]where \(a\), \(b\), and \(c\) are real numbers such that \(a \neq 0\).

The first derivative of \(f\) is given by

\[ f'(x) = 2ax + b \]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

\[ 2ax + b > 0 \] which may be written as \[ 2 a x > - b \]We now need to consider two cases and continue solving the inequality above.

We divide both sides of the inequality by \(2a\) and solve it to obtain

\[ x > -\frac{b}{2a} \]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 \((- \infty, -b/2a)\) and increasing over the interval \((-b/2a, + \infty)\).

We divide both sides of the inequality by \(2a\) but because \(a\) is less than 0, we need to change the symbol of inequality

\[ x \lt -\frac{b}{2a} \]We now analyze the sign of \(f'\) using the table below

The quadratic function with \(a \lt 0\) has a maximum point at \((-b/2a, f(-b/2a))\) and the function is increasing on the interval \((- \infty, -b/2a)\) and decreasing over the interval \((-b/2a, + \infty)\).

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 \(a \neq 0\).

The first derivative of \(f\) is given by

\[ f'(x) = 2a(x - h) \]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.

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 \((- \infty, h)\) and increasing over the interval \((h, + \infty)\).

We divide both sides of the inequality by \(a\) but we need to change the symbol of inequality because \(a\) is less than 0.

\[ x \lt h \]We analyze the sign of \(f'\) using the table below

The quadratic function with \(a \lt 0\) has a maximum point at \((h, k)\) and the function is increasing on the interval \((- \infty, h)\) and decreasing over the interval \((h, + \infty)\).

Find the extremum (minimum or maximum) of the quadratic function \(f\) given by

\[ f(x) = 2x^2 - 8x + 1 \]- We first find the derivative

\(f'(x) = 4x - 8\)

\(f'(x)\) changes sign at \(x = \frac{8}{4} = 2\). The leading coefficient \(a\) is positive hence \(f\) has a minimum at \((2, f(2)) = (2, -7)\) and \(f\) is decreasing on \((- \infty, 2)\) and increasing on \((2, + \infty)\). See graph below to confirm the result obtained by calculations.

Find the extremum (minimum or maximum) of the quadratic function \(f\) given by

\[ f(x) = - (x + 3)^2 + 1 \]- 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 \((- \infty, -3)\) and decreasing on \((-3, + \infty)\). See graph below of \(f\) below.

a) \(f(x) = x^2 + 6x\)

b) \(f(x) = -x^2 - 2x + 3\)

c) \(f(x) = x^2 - 5\)

d) \(f(x) = -(x - 4)^2 + 2\)

e) \(f(x) = -x^2\)

decreasing on \( (-\infty, -3) \)

increasing on \( (-3, +\infty) \)

b) maximum at at the point \( (-1, 4) \)

increasing on the interval \( (-\infty, -1) \)

decreasing on the interval \( (-1, +\infty) \)

c) minimum at at the point \( (0 , -5) \)

decreasing on the interval\( (-\infty, 0) \)

increasing on the interval \( (0, +\infty) \)

d) maximum at \( (4, 2) \)

increasing on the interval \( (-\infty, 4) \)

decreasing on the interval \( (4, +\infty) \)

e) maximum at \( (0, 0) \)

increasing on the interval \( (-\infty, 0) \)

decreasing on the interval \( (0, +\infty) \)