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Understand and use the unit circle.
a. 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 counterclockwise around the unit circle.
b. 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.
c. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

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Proving identities and formulas within the context of trigonometry.
a. Prove the Pythagorean identity 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.
b. Prove the addition and subtraction formulas for sine, cosine and tangent and use them to solve problems.

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

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Recognize that arithmetic and geometric sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

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Graph functions expressed symbolically and show key features of the graph, with and without using technology (computer, graphing calculator). ★
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b. Graph square root, cube root, and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
d. Graph exponential and logarithmic functions, showing intercepts and end behavior.

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Use arithmetic and geometric sequences to model situations and scenarios.
a. Use formulas (explicit and recursive) to generate terms for arithmetic and geometric sequences.
b. Write formulas to model arithmetic and geometric sequences and apply those formulas in realistic situations. ★ c. Translate between recursive and explicit formulas.

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Understand the relationship of radian measure of an angle to its arc length.

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Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents with the use of technology.

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Understand the effects of transformations on the graph of a function.
a. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x),f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.
b. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

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Find inverse functions.
a. Given the equation of an invertible function, find the inverse.
b. Verify by composition that one function is the inverse of another.
c. Read values of an inverse function from a graph or a table, given that the function has an inverse.
d. Produce an invertible function from a non-invertible function by restricting the domain.

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Write a function that describes a relationship between two quantities.★
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
b. Combine standard function types using arithmetic operations.
c. Compose functions.

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Distinguish between situations that can be modeled with linear functions and with exponential functions.
a. Recognize and justify that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

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Understand properties and key features of functions and the different way functions can be represented.
a. 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.
b. Use appropriate function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context.
c. 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.
d. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
e. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

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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).

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

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Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. Identify zeros, extreme values, and symmetry of the graph within the context of a quadratic function.
b. Use the properties of exponents to interpret expressions for exponential functions and classify the exponential function as representing growth or decay.

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Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

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Use graphs to represent functions.
a. Describe qualitatively the functional relationship between two quantities by analyzing a graph.
b. Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

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Understand average rate of change of a function over an interval.
a. Calculate and interpret the average rate of change of a function(presented symbolically or as a table) over a specified interval.
b. Estimate the rate of change from a graph. ★

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Understand properties of linear functions.
a. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line.
b. Identify and give examples of functions that are not linear.

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Construct a function to model a linear relationship between two quantities.
a. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph.
b. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values.

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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. ★

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Choose trigonometric functions to model periodic phenomena with specified period, midline and amplitude. ★

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Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

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Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.