Chapter 8: Graphs of Functions

8.1 Functions

A function is a relation where each element in the domain (input, $x$) is mapped to exactly one element in the codomain (output, $y$).

Types of Relations

• One-to-One: Represents a function (e.g., $y = x + 2$).
• Many-to-One: Represents a function (e.g., $y = x^2$).
• One-to-Many: Not a function.
• Many-to-Many: Not a function.

8.2 Graphs of Functions

A graph of a function is a visual representation of the relationship between an independent variable ($x$) and a dependent variable ($y$) plotted on a Cartesian coordinate plane.

Common Types of Functions & Shapes

• Linear Function: $y = ax + b$ — Straight line.
• Quadratic Function: $y = ax^2 + bx + c$ — Parabola (U-shape if $a > 0$, inverted U-shape if $a < 0$).
• Cubic Function: $y = ax^3 + c$ — Smooth curve with an 'S' bend.
• Reciprocal Function: $y = \frac{a}{x}$ — Hyperbola (two curved branches).

Constructing & Interpreting Graphs

1. Construct a table of values for given $x$-values.
2. Plot the coordinates $(x, y)$ using an appropriate scale on both axes.
3. Join the plotted points smoothly (or with a straight edge for linear functions).
4. Read values of $y$ for given values of $x$ (or vice versa) directly from the graph.