West Virginia

PRR Polynomial, Rational, and Radical Relationships

• Perform arithmetic operations with complex numbers.

  • 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.

    • Introduction to complex numbers (A2-K.1)

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

    • Add and subtract complex numbers (A2-K.2)
    • Multiply complex numbers (A2-K.4)
    • Powers of i (A2-K.8)

• Use complex numbers in polynomial identities and equations.

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

    • Solve a quadratic equation using square roots (A2-O.1)
    • Solve a quadratic equation by completing the square (A2-O.5)
    • Solve a quadratic equation using the quadratic formula (A2-O.6)
    • Using the discriminant (A2-O.8)

  • M.A2HS.4 Extend polynomial identities to the complex numbers.

    • Solve polynomial equations (A2-Q.1)
    • Complex conjugate theorem (A2-Q.5)
    • Conjugate root theorems (A2-Q.6)

  • M.A2HS.5 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

    • Fundamental Theorem of Algebra (A2-Q.8)

• Interpret the structure of expressions.

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

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

      • Polynomial vocabulary (A2-L.1)
      • Sort factors of single-variable expressions (A2-M.1)
      • Sort factors of multi-variable expressions (A2-M.2)
      • Interpret parts of quadratic expressions: word problems (A2-N.14)

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

      • Interpret parts of quadratic expressions: word problems (A2-N.14)

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

    • Pascal's triangle and the Binomial Theorem (A2-L.9)
    • Binomial Theorem I (A2-L.10)
    • Binomial Theorem II (A2-L.11)
    • Factor using a quadratic pattern (A2-M.6)
    • Factor by grouping (A2-M.7)
    • Factor sums and differences of cubes (A2-M.8)
    • Factor polynomials (A2-M.9)
    • Simplify radical expressions with variables I (A2-R.4)
    • Simplify radical expressions with variables II (A2-R.5)
    • Simplify radical expressions using conjugates (A2-R.11)
    • Convert between rational exponents and radicals (A2-T.2)
    • Simplify expressions involving rational exponents I (A2-T.6)
    • Simplify expressions involving rational exponents II (A2-T.7)
    • Simplify rational expressions (A2-X.4)

• Write expressions in equivalent forms to solve problems.

  • M.A2HS.8 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

    • Find the sum of a finite geometric series (A2-NN.4)

• Perform arithmetic operations on polynomials.

  • 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.

    • Add and subtract polynomials (A2-L.2)
    • Multiply polynomials (A2-L.3)

• Understand the relationship between zeros and factors of polynomials.

  • 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).

    • Evaluate polynomials using synthetic division (A2-L.7)
    • Write a quadratic function from its zeros (A2-N.10)
    • Find the roots of factored polynomials (A2-Q.2)
    • Write a polynomial from its roots (A2-Q.3)

  • 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.

    • Solve a quadratic equation using the zero product property (A2-O.2)
    • Find the roots of factored polynomials (A2-Q.2)
    • Match polynomials and graphs using zeros (A2-Q.10)
    • Use zeros to match quadratic functions and graphs (A1)
    • Use zeros to match quadratic functions and graphs (A2)

• Use polynomial identities to solve problems.

  • M.A2HS.12 Prove polynomial identities and use them to describe numerical relationships.

    • Factor sums and differences of cubes (A2-M.8)

  • 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.

    • Pascal's triangle (A2-L.8)
    • Pascal's triangle and the Binomial Theorem (A2-L.9)
    • Binomial Theorem I (A2-L.10)
    • Binomial Theorem II (A2-L.11)

• Rewrite rational expressions.

  • M.A2HS.14 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

    • Divide polynomials using long division (A2-L.4)
    • Simplify rational expressions (A2-X.4)

  • M.A2HS.15 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

    • Multiply and divide rational expressions (A2-X.5)
    • Add and subtract rational expressions (A2-X.6)
    • Simplify complex fractions (A2-X.7)

• Understand solving equations as a process of reasoning and explain the reasoning.

  • 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.

    • Solve radical equations (A2-S.4)
    • Solve rational equations (A2-X.8)

• Represent and solve equations and inequalities graphically.

  • 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.

    • Find solutions using a table (A2-A.7)
    • Approximate solutions using a table (A2-A.8)
    • Solve a system of equations by graphing (A2-E.2)
    • Solve a system of equations by graphing: word problems (A2-E.3)
    • Find the number of solutions to a system of equations (A2-E.4)

• Analyze functions using different representations.

  • 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.

    • Graph a quadratic function (A2-N.5)
    • Match quadratic functions and graphs (A2-N.7)
    • Match polynomials and graphs (A2-Q.11)

TF Trigonometric Functions

• Extend the domain of trigonometric functions using the unit circle.

  • M.A2HS.19 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

    • Convert between radians and degrees (A2-HH.1)
    • Radians and arc length (A2-HH.2)

  • M.A2HS.20 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.

    • Coterminal angles (A2-HH.5)
    • Reference angles (A2-HH.6)
    • Find trigonometric ratios using the unit circle (A2-II.6)

• Model periodic phenomena with trigonometric functions.

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

    • Find properties of sine functions (A2-KK.1)
    • Write equations of sine functions from graphs (A2-KK.2)
    • Write equations of sine functions using properties (A2-KK.3)
    • Find properties of cosine functions (A2-KK.6)
    • Write equations of cosine functions from graphs (A2-KK.7)
    • Write equations of cosine functions using properties (A2-KK.8)

• Prove and apply trigonometric identities.

  • 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.

    • Find trigonometric ratios using a Pythagorean or reciprocal identity (A2-LL.3)
    • Find trigonometric ratios using multiple identities (A2-LL.4)

MF Modeling with Functions

• Create equations that describe numbers or relationships.

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

    • Write inequalities from graphs (A2-C.2)
    • Solve quadratic equations: word problems (A2-O.7)
    • Write exponential functions: word problems (A2-Z.3)

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

    • Graph a linear function (A2-D.2)
    • Write the equation of a linear function (A2-D.3)
    • Graph an absolute value function (A2-H.3)
    • Graph a quadratic function (A2-N.5)
    • Write a quadratic function in vertex form from its vertex and another point (A2-N.11)
    • Write a quadratic function in factored form from its x-intercepts and another point (A2-N.12)
    • Write a quadratic function in standard form from its x-intercepts and another point (A2-N.)
    • Write a quadratic function from three points (A2-N.13)
    • Graph square root functions (A2-S.2)
    • Graph cube root functions (A2-S.3)
    • Write and solve direct variation equations (A2-Y.1)
    • Write and solve inverse variation equations (A2-Y.2)
    • Write joint and combined variation equations I (A2-Y.4)
    • Write joint and combined variation equations II (A2-Y.6)
    • Solve variation equations (A2-Y.7)
    • Write exponential functions: word problems (A2-Z.3)
    • Graph exponential functions (A2-Z.4)
    • Graph logarithmic functions (A2-BB.2)
    • Write equations of sine functions from graphs (A2-KK.2)
    • Write equations of sine functions using properties (A2-KK.3)
    • Graph sine functions (A2-KK.4)
    • Write equations of cosine functions from graphs (A2-KK.7)
    • Write equations of cosine functions using properties (A2-KK.8)
    • Graph cosine functions (A2-KK.9)
    • Write equations of sine and cosine functions from graphs (A2-KK.12)
    • Write equations of sine and cosine functions using properties (A2-KK.13)
    • Graph sine and cosine functions (A2-KK.14)

  • 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.)

    • Solve a system of equations by graphing: word problems (A2-E.3)
    • Solve a system of equations using substitution: word problems (A2-E.7)
    • Solve a system of equations using elimination: word problems (A2-E.9)
    • Solve a system of equations using any method: word problems (A2-E.11)
    • Write two-variable linear inequalities: word problems (A2-F.1)

  • 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.

    • Rearrange multi-variable equations (A2-B.4)

• Interpret functions that arise in applications in terms of a context.

  • 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.

    • Identify graphs: word problems (A2-A.6)
    • Find the slope of a linear function (A2-D.1)
    • Graph a linear function (A2-D.2)
    • Graph an absolute value function (A2-H.3)
    • Characteristics of quadratic functions: graphs (A2-N.1)
    • Characteristics of quadratic functions: equations (A2-N.2)
    • Graph a quadratic function (A2-N.5)
    • Match quadratic functions and graphs (A2-N.7)
    • Match polynomials and graphs (A2-Q.11)
    • Graph square root functions (A2-S.2)
    • Graph cube root functions (A2-S.3)
    • Rational functions: asymptotes and excluded values (A2-X.1)
    • Classify variation (A2-Y.3)
    • Find the constant of variation (A2-Y.5)
    • Graph exponential functions (A2-Z.4)
    • Match exponential functions and graphs (A2-Z.5)
    • Graph logarithmic functions (A2-BB.2)

  • 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.)

    • Domain and range (A2-A.1)
    • Domain and range of absolute value functions: graphs (A2-H.1)
    • Domain and range of absolute value functions: equations (A2-H.2)
    • Domain and range of quadratic functions: graphs (A2-N.8)
    • Domain and range of quadratic functions: equations (A2-N.9)
    • Domain and range of polynomials (A2-Q.12)
    • Domain and range of radical functions (A2-S.1)
    • Domain and range of exponential functions (A2-Z.1)
    • Domain and range of logarithmic functions (A2-BB.1)

  • 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.

    • Average rate of change (A2-A.9)
    • Find the slope of a linear function (A2-D.1)

• Analyze functions using different representations.

  • 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.

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

      • Graph an absolute value function (A2-H.3)
      • Graph piecewise-defined functions (A2-I.2)
      • Graph square root functions (A2-S.2)
      • Graph cube root functions (A2-S.3)

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

      • Graph exponential functions (A2-Z.4)
      • Match exponential functions and graphs (A2-Z.5)
      • Graph logarithmic functions (A2-BB.2)
      • Find properties of sine functions (A2-KK.1)
      • Graph sine functions (A2-KK.4)
      • Graph translations of sine functions (A2-KK.5)
      • Find properties of cosine functions (A2-KK.6)
      • Graph cosine functions (A2-KK.9)
      • Graph translations of cosine functions (A2-KK.10)
      • Graph sine and cosine functions (A2-KK.14)
      • Graph translations of sine and cosine functions (A2-KK.15)

  • M.A2HS.31 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

    • Characteristics of quadratic functions: graphs (A2-N.1)
    • Solve a quadratic equation by factoring (A2-O.3)
    • Solve a quadratic equation by completing the square (A2-O.5)
    • Evaluate exponential functions (A2-Z.2)
    • Match exponential functions and graphs (A2-Z.5)
    • Exponential functions over unit intervals (A2-Z.6)
    • Solve exponential equations by rewriting the base (A2-CC.1)

  • 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.)

• Build a function that models a relationship between two quantities.

  • 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.)

    • Add and subtract polynomials (A2-L.2)
    • Multiply polynomials (A2-L.3)
    • Add and subtract functions (A2-U.1)
    • Multiply functions (A2-U.2)
    • Divide functions (A2-U.3)
    • Multiply and divide rational expressions (A2-X.5)
    • Add and subtract rational expressions (A2-X.6)

• Build new functions from existing functions.

  • 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.

    • Even and odd functions (A2-Q.13)
    • Function transformation rules (A2-W.1)
    • Translations of functions (A2-W.2)
    • Reflections of functions (A2-W.3)
    • Dilations of functions (A2-W.4)
    • Transformations of functions (A2-W.5)
    • Describe function transformations (A2-W.6)

  • 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.)

    • Identify inverse functions (A2-V.1)
    • Find inverse functions and relations (A2-V.4)
    • Solve logarithmic equations I (A2-CC.6)
    • Solve logarithmic equations II (A2-CC.7)
    • Find the inverse of a quadratic function (A2)
    • Find the inverse of a cubic function (A2)
    • Find the inverse of a rational function (A2)

• Construct and compare linear, quadratic, and exponential models and solve problems.

  • 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.

    • Convert between exponential and logarithmic form: rational bases (A2-AA.1)
    • Convert between natural exponential and logarithmic form (A2-AA.2)
    • Convert between exponential and logarithmic form: all bases (A2-AA.3)
    • Evaluate logarithms (A2-AA.4)
    • Solve exponential equations using common logarithms (A2-CC.2)
    • Solve exponential equations using natural logarithms (A2-CC.3)

ICD Inferences and Conclusions from Data

• Summarize, represent, and interpret data on a single count or measurement variable.

  • M.A2HS.37 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

    • Find probabilities using the normal distribution I (A2-QQ.2)
    • Find z-values (A2-QQ.4)
    • Find values of normal variables (A2-QQ.5)
    • Distributions of sample means (A2-QQ.6)

• Understand and evaluate random processes underlying statistical experiments.

  • M.A2HS.38 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

    • Identify biased samples (A2-RR.1)

  • 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?)

• Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

  • M.A2HS.40 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    • Experiment design (A2-RR.8)

  • 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.

    • Find confidence intervals for population means (A2-RR.5)
    • Find confidence intervals for population proportions (A2-RR.6)

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

    • Analyze the results of an experiment using simulations (A2-RR.9)

  • M.A2HS.43 Evaluate reports based on data.

    • Interpret confidence intervals for population means (A2-RR.7)
    • Analyze the results of an experiment using simulations (A2-RR.9)

• Use probability to evaluate outcomes of decisions.

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

  • 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).

    • Choose the better bet (A2-PP.10)