### PRR: Polynomial, Rational, and Radical Relationships

#### (Framing Text): Perform arithmetic operations with complex numbers.

PRR.M.A2HS.1: Know there is a complex number i such that i² = −1, and every complex number has the form a + bi with a and b real.

PRR.M.A2HS.2: Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

#### (Framing Text): Use complex numbers in polynomial identities and equations.

PRR.M.A2HS.3: Solve quadratic equations with real coefficients that have complex solutions. Instructional Note: Limit to polynomials with real coefficients.

#### (Framing Text): Interpret the structure of expressions.

PRR.M.A2HS.6: Interpret expressions that represent a quantity in terms of its context.

PRR.M.A2HS.6.a: Interpret parts of an expression, such as terms, factors, and coefficients.

PRR.M.A2HS.6.b: Interpret complicated expressions by viewing one or more of their parts as a single entity.

PRR.M.A2HS.7: Use the structure of an expression to identify ways to rewrite it.

#### (Framing Text): Perform arithmetic operations on polynomials.

PRR.M.A2HS.9: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

#### (Framing Text): Understand the relationship between zeros and factors of polynomials.

PRR.M.A2HS.10: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

PRR.M.A2HS.11: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

#### (Framing Text): Use polynomial identities to solve problems.

PRR.M.A2HS.13: Know and apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.

#### (Framing Text): Understand solving equations as a process of reasoning and explain the reasoning.

PRR.M.A2HS.16: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Instructional Note: Extend to simple rational and radical equations.

#### (Framing Text): Represent and solve equations and inequalities graphically.

PRR.M.A2HS.17: 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.

#### (Framing Text): Analyze functions using different representations.

PRR.M.A2HS.18: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

### TF: Trigonometric Functions

#### (Framing Text): Model periodic phenomena with trigonometric functions.

TF.M.A2HS.21: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

#### (Framing Text): Prove and apply trigonometric identities.

TF.M.A2HS.22: 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.

### MF: Modeling with Functions

#### (Framing Text): Create equations that describe numbers or relationships.

MF.M.A2HS.23: Create equations and inequalities in one variable and use them to solve problems.

MF.M.A2HS.24: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

MF.M.A2HS.25: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. (e.g., Represent inequalities describing nutritional and cost constraints on combinations of different foods.)

MF.M.A2HS.26: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.) While functions will often be linear, exponential, or quadratic the types of problems should draw from more complex situations than those addressed in Algebra I.

#### (Framing Text): Interpret functions that arise in applications in terms of a context.

MF.M.A2HS.27: 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.

MF.M.A2HS.28: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (e.g., If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.)

MF.M.A2HS.29: 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.

#### (Framing Text): Analyze functions using different representations.

MF.M.A2HS.30: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

MF.M.A2HS.30.a: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

MF.M.A2HS.30.b: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

MF.M.A2HS.32: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.)

#### (Framing Text): Build a function that models a relationship between two quantities.

MF.M.A2HS.33: Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.)

#### (Framing Text): Build new functions from existing functions.

MF.M.A2HS.34: 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. 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.

MF.M.A2HS.35: Find inverse functions. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. (e.g., f(x) = 2 x³ or f(x) = (x+1)/(x-1) for x ≠ 1.)

#### (Framing Text): Construct and compare linear, quadratic, and exponential models and solve problems.

MF.M.A2HS.36: For exponential models, express as a logarithm the solution to a b to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

### ICD: Inferences and Conclusions from Data

#### (Framing Text): Understand and evaluate random processes underlying statistical experiments.

ICD.M.A2HS.39: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. (e.g., A model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?)

#### (Framing Text): Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

ICD.M.A2HS.41: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

ICD.M.A2HS.42: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

#### (Framing Text): Use probability to evaluate outcomes of decisions.

ICD.M.A2HS.44: Use probabilities to make fair decisions (e.g., drawing by lots or using a random number generator).

ICD.M.A2HS.45: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, and/or pulling a hockey goalie at the end of a game).

Correlation last revised: 9/16/2020

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