How to Graph, Evaluate, and Write Piecewise Functions

Miacademy & MiaPrep Learning Channel

This comprehensive math tutorial demystifies piecewise functions by connecting abstract algebra concepts to real-world scenarios. The video begins with a relatable "Buy More, Save More" retail example to intuitively explain why a single function might require different rules for different input values. It progresses into formal mathematical definitions, demonstrating how to evaluate function values based on specific domain intervals and how to handle inequalities. The core of the lesson provides a step-by-step walkthrough of graphing a complex piecewise function that includes linear, absolute value, quadratic, and constant components. It pays special attention to common student stumbling blocks, such as distinguishing between open and closed circles at boundary points and handling gaps in the domain. The video also teaches students how to determine the overall range of a piecewise function by analyzing the output intervals of its individual parts. Finally, the video scaffolds the advanced skill of "reverse engineering" a function—writing the algebraic equations when given a graph. By reinforcing prior knowledge of slope-intercept form and function transformations, this resource serves as an excellent primary lesson for Algebra 2 students or a robust review for Pre-Calculus students encountering these "Frankenstein" functions.