# Functions

## • Interpret key features of graphs in terms of the quantities represented (F-IF.B.4$^\star$). • Sketch graphs showing key features of the graph by hand and using technology (F-IF.C.7$^\star$). • Understand that a function from one set (the domain) to another set (the range) assigns to each element of the domain exactly one element of the range (F-IF.A.1). • Use function notation (F-IF.A.2). • Interpret statements that use function notation in various contexts (F-IF.A.2). • Work with graphs of piecewise-defined functions, including step functions (F-IF.C.7b$^\star$). • Relate the domain of a function to its graph (F-IF.B.5$^\star$). • Relate the domain of a function to the quantitative relationship it describes (F-IF.B.5$^\star$). • Calculate and interpret the average rate of change of a function over a specified interval (F-IF.B.6$^\star$). • Estimate the average rate of change of a function from its graph (F-IF.B.6$^\star$). • Solve for x such that f(x) = c, when f is a linear function (F-BF.B.4a). • Write an expression for the inverse of a linear function (F-BF.B.4a).

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.

## Sections

A1.4.0 Pre-unit diagnostic assessment

#### Summary

Assess students’ ability to
• identify functions and non-functions (8.F.A.1);
• identify linear and nonlinear functions (8.F.A.3);
• write an equation for the corresponding linear function when given two points on a line (8.F.B.4);
• find and interpret the rate of change when given a linear function that models a situation (8.F.B.4);
• interpret the graph of a function in terms of the situation it models (8.F.B.4).

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A1.4.1 Graphing and functions

#### Summary

• Sketch graphs showing key features (F-IF.B.4$^\star$, F-IF.C.7$^\star$).
• Interpret key features of graphs in terms of the quantities represented (F-IF.B.4$^\star$, F-IF.C.7$^\star$).

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A1.4.2 Introducing function notation

#### Summary

Understand that a function assigns to each element in the domain exactly one element of the range (F-IF.A.1).

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A1.4.3 Interpreting function notation in context

#### Summary

Interpret statements that use function notation in terms of the quantities represented (F-IF.A.2).

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A1.4.4 Mid-unit assessment

#### Summary

Assess students’ ability to
• use function notation to represent points on a graph and to describe features of a graph (F-IF.A.1);
• understand function notation and how to interpret statements in function notation in terms of a context (F-IF.A.2).

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A1.4.5 Domain, range, piecewise-defined functions

#### Summary

• Use the notions of domain and range (F-IF.A.1, F-IF.B.5$^\star$).
• Interpret a graph of a piecewise-defined function (F-IF.C.7b$^\star$).
• Graph step functions (F-IF.C.7b$^\star$).

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A1.4.6 Interpreting graphs in context

#### Summary

• Interpret key features of graphs in terms of the quantities represented (F-IF.B.4$^\star$).
• Sketch graphs showing key features (F-IF.B.4$^\star$).
• Relate the domain of a function to its graph (F-IF.B.5$^\star$).
• Relate the domain to the quantitative relationship it describes (F-IF.B.5$^\star$).

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A1.4.7 Average rate of change

#### Summary

• Calculate the average rate of change of a function over a specified interval (F-IF.B.6$^\star$)
• Interpret the average rate of change of a function over a specified interval (F-IF.B.6$^\star$).

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A1.4.8 Inverse functions

#### Summary

• Solve for $x$ such that $f(x) = c$, when $f$ is a linear function (F-BF.B.4a).
• Write an expression for an inverse (F-BF.4a).

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A1.4.9 Summative assessment

#### Summary

Assess students’ ability to
• identify the domain and range of a given function (F-IF.A.1);
• create a step function that represents pairs of given values (F-IF.C.7b$^\star$);
• interpret statements that use function notation in a context (f-IF.A.2);
• identify key features of graphs (F-IF.B.4$^\star$);
• identify referents of expressions that use function notation in a given graph (F-IF.B.4$^\star$);
• identify and relate the domain and range of a function in a given context (F-IF.B.5);
• calculate the average rate of change of a function over a specified interval (F-IF.B.6$^\star$);
• interpret the average rate of change of a function in terms of the quantities represented (F-IF.B.6$^\star$).

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