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
\[ f(x) = 2x^2 - 8x + 1 \]Find the extremum (minimum or maximum) of the quadratic function \(f\) given by
\[ f(x) = - (x + 3)^2 + 1 \]
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) \(f(x) = -x^2\)