Graph Intercepts

Definition

Intercepts are the points where the graph of an equation or function intersects the coordinate axes.

Why It Matters

Intercepts are the ‘low-hanging fruit’ of function analysis; they provide immediate insights into where a system ‘zeros out’ or starts (its initial value), allowing for quick sketching and simplified problem-solving.

Core Concepts

• $x$-intercepts: Set $y=0$ and solve for $x$.
  • How to read: “Set y equal to zero and solve for x.”
  • Meaning: Points where the graph crosses the $x$-axis — zeros or roots of the function.
• $y$-intercepts: Set $x=0$ and solve for $y$.
  • How to read: “Set x equal to zero and solve for y.”
  • Meaning: Where the graph crosses the $y$-axis — evaluate $f(0)$ if defined. Often corresponds to the initial state of a system.

Connected Concepts

• Graph Symmetry
• Graphs of Functions
• Polynomial Functions, Polynomial Graphs