Functions

The story before this unit:

In grade 8, students are first introduced to the notion of functions. They understand a function as a rule that assigns to each input exactly one output 8.F.A.1. The main focus is on

• linear functions and their representations (equations, graphs, tables, or verbal descriptions);
• understanding that the equation $y = mx + b$ defines a linear function, whose graph is a line;
• modeling a linear relationship by a function, determining rate of change and initial value and interpreting them in terms of a situation modeled by the function and in terms of its graph or a table of values;
• describing qualitatively a functional relationship between two quantities by analyzing a graph, e.g., where it is increasing or decreasing, linear or nonlinear.

 

The part of the story happening in this unit:

In this unit, students begin to use formal notation for functions, writing equations such as $f(x) = 2x + 3$ to describe a function. Students develop the understanding that the input/output relationship is a correspondence between two sets, and use the terms domain and range to describe them. They develop fluency with function notation and its use in describing qualitative features of the graph of a function by first interpreting, then writing expressions, equations, and inequalities such as $f(x + 2)$, $f(a) = 2$, $f(x) > 2$, and $f(x) > g(x).$

Students expand their repertoires of functions, working with piecewise-defined functions, including step functions.

Building on their experiences with rate of change and slope from grade 8, students examine the behavior of non-linear functions. They describe key aspects of their graphs, and calculate and interpret average rates of change over specified intervals.

Students’ work with domain and range provides a basis for understanding when a function has an inverse. They examine simple functions that do and do not have inverses and write expressions for inverses of linear functions, but do not draw general conclusions about when a function has an inverse.

The story after this unit:

Functions are a unifying theme in high school mathematics. In statistics, functions play especially prominent supporting roles as lines of best fit for bivariate statistics and the normal distribution. In geometry, transformations are viewed as functions sending points in the plane to points in the plane, and ratios of sides of right triangles lead to the trigonometric functions on acute angles. In algebra, students study systems of linear equations and inequalities, exponential functions and geometric sequences, quadratic functions, and rational functions, polynomials, and logarithms.