Functions

Build an understanding that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range by recognizing that: • if f is a function and x is an element of its domain, then 𝑓(𝑥) denotes the output of f corresponding to the input x. • the graph of 𝑓 is the graph of the equation 𝑦 = 𝑓(𝑥).

Use function notation to evaluate linear, quadratic, and exponential functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Recognize that recursively and explicitly defined sequences are functions whose domain is a subset of the integers, the terms of an arithmetic sequence are a subset of the range of a linear function, and the terms of a geometric sequence are a subset of the range of an exponential function.

Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; and maximums and minimums.

Interpret a function in terms of the context by relating its domain and range to its graph and, where applicable, to the quantitative relationship it describes.

Calculate and interpret the average rate of change over a specified interval for a function presented numerically, graphically, and/or symbolically.

Analyze linear, exponential, and quadratic functions by generating different representations, by hand in simple cases and using technology for more complicated cases, to show key features, including: domain and range; rate of change; intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; and end behavior.

Use equivalent expressions to reveal and explain different properties of a function.

Rewrite a quadratic function to reveal and explain different key features of the function.

Interpret and explain growth and decay rates for an exponential function.

Compare key features of two functions (linear, quadratic, or exponential) each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).

Write a function that describes a relationship between two quantities.

Build linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two ordered pairs (include reading these from a table).

Build a function that models a relationship between two quantities by combining linear, exponential, or quadratic functions with addition and subtraction or two linear functions with multiplication.

Translate between explicit and recursive forms of arithmetic and geometric sequences and use both to model situations.

Identify situations that can be modeled with linear and exponential functions, and justify the most appropriate model for a situation based on the rate of change over equal intervals.

Compare the end behavior of linear, exponential, and quadratic functions using graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.

Interpret the parameters 𝑎 and 𝑏 in a linear function 𝑓(𝑥) = 𝑎𝑥 + 𝑏 or an exponential function 𝑔(𝑥) = 𝑎𝑏 𝑥 in terms of a context.