High School — Functions

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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).F.IF.1

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.F.IF.2

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.F.IF.3

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 the following: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.F.IF.4

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.F.IF.5

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.F.IF.6

Graph functions expressed symbolically and indicate key features of the graph, by hand in simple cases and using technology for more complicated cases. Include applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate.F.IF.7

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.F.IF.8

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).F.IF.9

Write a function that describes a relationship between two quantities.F.BF.1

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.F.BF.2

Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.F.BF.3

Find inverse functions.F.BF.4

Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.(+)F.BF.5

Distinguish between situations that can be modeled with linear functions and with exponential functions.F.LE.1

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).F.LE.2

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.F.LE.3

For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.F.LE.4

Interpret the parameters in a linear or exponential function in terms of a context.F.LE.5

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.F.TF.1

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counter-clockwise around the unit circle.F.TF.2

Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π − x, π + x, and 2π − x in terms of their values for x, where x is any real number.(+)F.TF.3

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.(+)F.TF.4

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.F.TF.5

Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.(+)F.TF.6

Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.(+)F.T.7

Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1, and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.F.TF.8

Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems.(+)F.TF.9