### AR.Math.Content.HSF.IF: Interpreting Functions

#### AR.Math.Content.HSF.IF.A: Understand the concept of a function and use function notation.

AR.Math.Content.HSF.IF.A.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. Understand 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. Understand that the graph of 𝑓𝑓 is the graph of the equation 𝑦 = (𝑥).

AR.Math.Content.HSF.IF.A.2: In terms of a real-world context: use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation.

#### AR.Math.Content.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

AR.Math.Content.HSF.IF.B.4: 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.

AR.Math.Content.HSF.IF.B.5: Relate the domain of a function to its graph. Relate the domain of a function to the quantitative relationship it describes.

AR.Math.Content.HSF.IF.B.6: Calculate and interpret the average rate of change of a function (presented algebraically or as a table) over a specified interval. Estimate the rate of change from a graph.

#### AR.Math.Content.HSF.IF.C: Analyze functions using different representations.

AR.Math.Content.HSF.IF.C.7: Graph functions expressed algebraically and show key features of the graph, with and without technology. Graph linear and quadratic functions and, when applicable, show intercepts, maxima, and minima. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph exponential and logarithmic functions, showing intercepts and end behavior. Graph trigonometric functions, showing period, midline, and amplitude.

AR.Math.Content.HSF.IF.C.8: Write expressions for functions in different but equivalent forms to reveal key features of the function. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values (vertex), and symmetry of the graph, and interpret these in terms of a context. Use the properties of exponents to interpret expressions for exponential functions.

### AR.Math.Content.HSF.BF: Building Functions

#### AR.Math.Content.HSF.BF.A: Build a function that models a relationship between two quantities.

AR.Math.Content.HSF.BF.A.1: Write a function that describes a relationship between two quantities. From a context, determine an explicit expression, a recursive process, or steps for calculation. Combine standard function types using arithmetic operations. (e.g., given that f(x) and g(x) are functions developed from a context, find (f + g)(x), (f – g)(x), (fg)(x), (f/g)(x), and any combination thereof, given 𝑔 (𝑥) ≠ 0.) Compose functions.

AR.Math.Content.HSF.BF.A.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, and translate between the two forms. Use arithmetic and geometric sequences to model situations.

#### AR.Math.Content.HSF.BF.B: Build new functions from existing functions.

AR.Math.Content.HSF.BF.B.3: 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 (𝑘, a constant both positive and negative); Find the value of 𝑘 given the graphs of the transformed functions. Experiment with multiple transformations and illustrate an explanation of the effects on the graph with or without technology.

AR.Math.Content.HSF.BF.B.4: Find inverse functions. Solve an equation of the form 𝑦 = 𝑓(𝑥) for a simple function f that has an inverse and write an expression for the inverse. Verify by composition that one function is the inverse of another. (Algebra II) Read values of an inverse function from a graph or a table, given that the function has an inverse. (Algebra II) Produce an invertible function from a non-invertible function by restricting the domain.

AR.Math.Content.HSF.BF.B.5: Understand the inverse relationship between exponents and logarithms. Use the inverse relationship between exponents and logarithms to solve problems.

### AR.Math.Content.HSF.LE: Linear, Quadratic, and Exponential Models

#### AR.Math.Content.HSF.LE.A: Construct and compare linear, quadratic, and exponential models and solve problems.

AR.Math.Content.HSF.LE.A.1: Distinguish between situations that can be modeled with linear functions and with exponential functions. Show that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

AR.Math.Content.HSF.LE.A.2: Construct linear and exponential equations, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

AR.Math.Content.HSF.LE.A.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or any polynomial function.

AR.Math.Content.HSF.LE.A.4: Express exponential models as logarithms. Express logarithmic models as exponentials. Use properties of logarithms to simplify and evaluate logarithmic expressions (expanding and/or condensing logarithms as appropriate). Evaluate logarithms with or without technology.

#### AR.Math.Content.HSF.LE.B: Interpret expressions for functions in terms of the situation they model.

AR.Math.Content.HSF.BF.B.5: In terms of a context, interpret the parameters (rates of growth or decay, domain and range restrictions where applicable, etc.) in a function.

### AR.Math.Content.HSF.TF: Trigonometric Functions

#### AR.Math.Content.HSF.TF.A: Extend the domain of trigonometric functions using the unit circle.

AR.Math.Content.HSF.TF.A.2: 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 around the unit circle.

AR.Math.Content.HSF.TF.A.3: Use special right triangles to determine geometrically the exact values of sine, cosine, tangent for π/3, π/4, π/6, and π/2. 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 exact values for 𝑥, where 𝑥 is any real number.

#### AR.Math.Content.HSF.TF.B: Model periodic phenomena with trigonometric functions.

AR.Math.Content.HSF.TF.B.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

#### AR.Math.Content.HSF.TF.C: Prove and apply trigonometric identities.

AR.Math.Content.HSF.TF.C.8: Develop the Pythagorean identity, sin²(𝜃) + cos²(𝜃) = 1. Given sin(𝜃), cos(𝜃), or tan(𝜃) and the quadrant of the angle, use the Pythagorean identity to find the remaining trigonometric functions.

AR.Math.Content.HSF.TF.C.9: Develop the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Correlation last revised: 9/16/2020

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