Differentiation is used to analyze key properties of quadratic functions such as intervals of increase and decrease, local maxima, and local minima.
A - Quadratic Function in General Form
Quadratic functions in their general form are written as:
where \(a\), \(b\), and \(c\) are real numbers with \(a \neq 0\).
The first derivative of \(f\) is:
\[ f'(x) = 2 a x + b \]To determine the maximum or minimum points and intervals of increase or decrease, analyze the sign of \(f'(x)\). The derivative is positive if:
\[ 2 a x + b > 0 \quad \Rightarrow \quad 2 a x > -b \]We consider two cases based on the sign of \(a\):
Case 1: \(a > 0\)
Dividing both sides of the inequality by \(2a > 0\), we get:
\[ x > -\dfrac{b}{2a} \]The table below summarizes the sign of \(f'(x)\) and whether \(f\) is increasing or decreasing:
Thus, the quadratic function has a minimum at \(\left(-\dfrac{b}{2a}, f\left(-\dfrac{b}{2a}\right)\right)\), decreases on \((-\infty, -\dfrac{b}{2a})\), and increases on \(\left(-\dfrac{b}{2a}, +\infty\right)\).
Case 2: \(a < 0\)
Dividing both sides by \(2a < 0\) reverses the inequality:
\[ x < -\dfrac{b}{2a} \]The sign of \(f'(x)\) is analyzed using the following table:
In this case, the quadratic function has a maximum at \(\left(-\dfrac{b}{2a}, f\left(-\dfrac{b}{2a}\right)\right)\), increases on \((-\infty, -\dfrac{b}{2a})\), and decreases on \(\left(-\dfrac{b}{2a}, +\infty\right)\).
B - Quadratic Function in Vertex Form
Quadratic functions in vertex form are written as:
where \(a \neq 0\), and \(h, k\) are real numbers.
The derivative is:
\[ f'(x) = 2 a (x - h) \]We analyze the sign of \(f'(x)\) by considering:
\[ a (x - h) > 0 \]Case 1: \(a > 0\)
Dividing both sides by \(a > 0\):
\[ x > h \]Use the following table to analyze the sign:
Therefore, the quadratic has a minimum at \((h, k)\), decreases on \((-\infty, h)\), and increases on \((h, +\infty)\).
Case 2: \(a < 0\)
Dividing both sides by \(a < 0\) reverses the inequality:
\[ x < h \]The sign of \(f'(x)\) is analyzed using:
Here, the quadratic has a maximum at \((h, k)\), increases on \((-\infty, h)\), and decreases on \((h, +\infty)\).
Worked Examples
Example 1: Find the extremum of \( f(x) = 2x^2 - 8x + 1 \)
First, find the derivative:
\[ f'(x) = 4x - 8 \]The derivative changes sign at \(x = \dfrac{8}{4} = 2\). Since \(a = 2 > 0\), \(f\) has a minimum at \((2, f(2)) = (2, -7)\).
Function behavior:
- Decreasing on \((-\infty, 2)\)
- Increasing on \((2, +\infty)\)
Example 2: Find the extremum of \( f(x) = - (x + 3)^2 + 1 \)
The derivative is:
\[ f'(x) = -2(x + 3) \]The derivative changes sign at \(x = -3\). Since \(a = -1 < 0\), \(f\) has a maximum at \((-3, 1)\).
Function behavior:
- Increasing on \((-\infty, -3)\)
- Decreasing on \((-3, +\infty)\)
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:
- \( f(x) = x^2 + 6x \)
- \( f(x) = -x^2 - 2x + 3 \)
- \( f(x) = x^2 - 5 \)
- \( f(x) = -(x - 4)^2 + 2 \)
- \( f(x) = -x^2 \)
Answers to the Exercises
- a) Minimum at \((-3, -9)\). Decreasing on \((-\infty, -3)\), increasing on \((-3, +\infty)\).
- b) Maximum at \((-1, 4)\). Increasing on \((-\infty, -1)\), decreasing on \((-1, +\infty)\).
- c) Minimum at \((0, -5)\). Decreasing on \((-\infty, 0)\), increasing on \((0, +\infty)\).
- d) Maximum at \((4, 2)\). Increasing on \((-\infty, 4)\), decreasing on \((4, +\infty)\).
- e) Maximum at \((0, 0)\). Increasing on \((-\infty, 0)\), decreasing on \((0, +\infty)\).