Growth & Decline

Understanding where a function increases or decreases is essential for analyzing its behavior and finding extreme values.

Increasing and Decreasing Functions

📐Definition — Increasing and Decreasing

A function $f$ is increasing on an interval $[a, b]$ if for any $x_1, x_2 \in [a, b]$ :

$x_1 < x_2 \implies f(x_1) < f(x_2)$

A function $f$ is decreasing on an interval $[a, b]$ if:

$x_1 < x_2 \implies f(x_1) > f(x_2)$

In other words, an increasing function “goes up” as you move to the right, and a decreasing function “goes down.”

Examples of Monotone Behavior

$f(x) = x^2$ : decreasing on $(-\infty, 0]$ and increasing on $[0, +\infty)$ .
$f(x) = x^3$ : increasing on all of $\mathbb{R}$ .
$f(x) = |x|$ : decreasing on $(-\infty, 0]$ and increasing on $[0, +\infty)$ .

A function can be neither increasing nor decreasing overall, yet still have monotone intervals. The function $f(x) = \sin x$ is a classic example — it increases on $[-\pi/2, \pi/2]$ and decreases on $[\pi/2, 3\pi/2]$ , repeating periodically.

Maximum and Minimum Values

A global maximum is the largest value the function attains on its entire domain. A global minimum is the smallest. A local maximum is a point where the function is larger than at all nearby points (and similarly for a local minimum).

⚡Theorem — Vertex of a Parabola

The quadratic function $f(x) = ax^2 + bx + c$ has its vertex at $x = -\dfrac{b}{2a}$ .

If $a > 0$ , the vertex is a global minimum.
If $a < 0$ , the vertex is a global maximum.

Worked Example

✎Example — Analyzing Growth and Decline

Analyze the function $f(x) = x^2 - 4x + 3$ .

Solution. Complete the square:

$f(x) = (x - 2)^2 - 1$

The vertex is at $(2, -1)$ .

Decreasing on $(-\infty, 2]$ : as $x$ increases toward $2$ , $(x-2)^2$ shrinks, so $f(x)$ decreases.
Increasing on $[2, +\infty)$ : as $x$ moves past $2$ , $(x-2)^2$ grows, so $f(x)$ increases.
Global minimum: $f(2) = -1$ . There is no global maximum (the function grows without bound).

Finding Intervals from a Graph

To identify where a function increases or decreases from its graph:

Locate all local maxima and minima (peaks and valleys).
Between consecutive extrema, the function is either entirely increasing or entirely decreasing.
Write the intervals using the $x$ -coordinates of these extrema.

✏Exercise

The function $g(x) = -x^2 + 6x - 5$ is a downward-opening parabola. Find the vertex and state the intervals of increase and decrease.