Graphs of Functions

Graphs of functions provide a visual representation of how output values change in response to varying inputs. Each point on the graph corresponds to an ordered pair, where the x-coordinate (independent variable) determines the horizontal position and the y-coordinate (dependent variable) determines the vertical position. Linear functions like y = x give a straight line, indicating a constant rate of change.

Nonlinear functions display more complex behaviors. Even power functions generate U-shaped curves that are symmetric about the y-axis, with non-negative outputs for both positive and negative x-values. Odd power functions form S-shaped curves that pass through the origin and extend in opposite directions, showing symmetry about the origin.

Reciprocal functions create two distinct curves that approach the coordinate axes without touching them. These axes serve as asymptotes—lines the graph approaches indefinitely but never intersects.

Function graphs are widely used in real-world applications. In physics and engineering, they model motion, forces, and systems. In economics, functions represent cost, revenue, and market behavior. In biology, they describe growth, decay, and reaction rates. Understanding these graphs enables accurate interpretation and prediction across diverse scientific and analytical domains.