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Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

Understand 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. 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).

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Distinguish between situations that can be modeled with linear functions and with exponential functions. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Informally assess the fit of a function by plotting and analyzing residuals.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Determine an explicit expression, a recursive process, or steps for calculation from a context. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Interpret the parameters in a linear or exponential function in terms of a context.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Graph functions expressed algebraically and show key features of the graph both by hand and by using technology. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.