Graphs of Functions: Properties

Definition

The properties of a function’s graph describe its visual and analytical behavior, including symmetry (even/odd), direction (increasing/decreasing), and critical points (extrema).

Why It Matters

The properties of a graph—increasing, decreasing, even, or odd—are the discrete ‘signals’ that describe a system’s evolution; mastering them allows you to read the ‘story’ of a function at a glance, identifying peaks, valleys, and trends without exhaustive calculation.

Core Concepts

Symmetry:
- Even Functions: $f(-x) = f(x)$ $f (- x) = f (x)$ . Symmetric about the $y$ $y$ -axis.
  - How to read: “The function f of negative x is equal to f of x.”
  - Meaning: Negating the input leaves the output unchanged — mirror symmetry across the $y$ -axis.
- Odd Functions: $f(-x) = -f(x)$ $f (- x) = - f (x)$ . Symmetric about the origin.
  - How to read: “The function f of negative x is equal to negative f of x.”
  - Meaning: Negating the input negates the output — 180° rotational symmetry about the origin.
Monotonicity (Direction):
- Increasing: $f(x_1) < f(x_2)$ $f (x_{1}) < f (x_{2})$ whenever $x_1 < x_2$ $x_{1} < x_{2}$ . The graph rises from left to right.
  - How to read: “The value f of x one is less than f of x two whenever x one is less than x two.”
  - Meaning: The function preserves order — graph climbs as $x$ increases.
- Decreasing: $f(x_1) > f(x_2)$ $f (x_{1}) > f (x_{2})$ whenever $x_1 < x_2$ $x_{1} < x_{2}$ . The graph falls from left to right.
  - How to read: “The value f of x one is greater than f of x two whenever x one is less than x two.”
  - Meaning: The function reverses order — graph descends as $x$ increases.
Local Extrema:
- Local Maximum: A point $(c, f(c))$ $(c, f (c))$ where $f(c) \ge f(x)$ $f (c) \geq f (x)$ for all $x$ $x$ in a nearby interval.
  - How to read: “The value f of c is greater than or equal to f of x for all x in a nearby interval.”
  - Meaning: A local peak — highest point in a small neighborhood.
- Local Minimum: A point $(c, f(c))$ $(c, f (c))$ where $f(c) \le f(x)$ $f (c) \leq f (x)$ for all $x$ $x$ in a nearby interval.
  - How to read: “The value f of c is less than or equal to f of x for all x in a nearby interval.”
  - Meaning: A local valley — lowest point in a small neighborhood.
Average Rate of Change: The slope of the secant line between two points $(a, f(a))$ and $(b, f(b))$ : $\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$
- How to read: “The ratio delta y divided by delta x is equal to the quantity f of b minus f of a, all over the quantity b minus a.”
- Meaning: Average slope between two points — “average speed” over the interval $[a,b].$

Graphs of Functions: Properties

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes