Unit 4: Functions

Definition of a Function: A function is a specific mathematical rule that assigns exactly one unique output to each input.

Representational Forms: Functions can be represented, translated, and analyzed across four distinct environments: algebraic equations (y=mx+b), coordinate graphs, input-output tables, and verbal descriptions.

Function Properties & Behavior: The independent variable corresponds to the input (domain), and the dependent variable corresponds to the output (range).

Linear vs. Nonlinear Dynamics: Linear functions possess a constant rate of change and graph as a straight line, whereas nonlinear functions do not have a constant rate of change.

Qualitative Features: Graphs can be read contextually from left to right to identify intervals where a function is qualitatively increasing, decreasing, or constant.

Test for Functions: Determine whether a mapping diagram, table of values, or coordinate graph represents a function (utilizing the vertical line test for graphs and checking for repeating inputs with differing outputs in tables).

Translate Representations: Convert a functional relationship seamlessly between equations, tables, graphs, and verbal situations.

Calculate and Compare Rates: Find and compare the rates of change (slopes) and initial values (y-intercepts) of two different functions, even when they are displayed in two different formats (e.g., comparing a table to an equation).

Construct Linear Models: Write a linear function equation (y=mx+b) to model a real-world relationship between two changing quantities.

Sketch and Describe Qualitative Graphs: Graph a story qualitatively by analyzing verbal descriptions (e.g., speed over time, distance from home) to show intervals of increasing, decreasing, or constant behavior.

• Function Classification Task: Correcting a set of "non-functions" and explaining why they fail the vertical line test (single output rule).

• Comparative Performance Task: Analyzing two different cell phone plans—one given as a table and one as an equation—to find the initial value and rate of change for both.

• Graph Storytelling: Students use a graph based on a verbal description of a trip (e.g., stopping for gas, driving on a highway) and label parts as increasing, decreasing, or constant.