M.A2HS: High School Algebra II

High School Algebra II

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High School Algebra II extends algebraic and function knowledge into complex numbers, polynomial and rational expressions, logarithms, and trigonometric functions. Students factor higher-degree polynomials, apply the Remainder and Binomial Theorems, and work with rational exponents and radical equations. Statistical inference, probability distributions, and trigonometric identities complete the course.

M.A2HS.1Know there is a complex number i such that i^2 = −1, and every complex number has the form a + bi with a and b real.M.A2HS.10Know 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).M.A2HS.11Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.M.A2HS.12Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2 = (x^2 – y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.M.A2HS.13Know 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.M.A2HS.14Rewrite 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.M.A2HS.15Understand 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.M.A2HS.16Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.M.A2HS.17Explain 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.M.A2HS.18Graph 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.M.A2HS.19Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.M.A2HS.2Use the relation i^2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.M.A2HS.20Explain 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.M.A2HS.21Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.M.A2HS.22Prove the Pythagorean identity sin^2(θ) + cos^2(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.M.A2HS.23Create equations and inequalities in one variable and use them to solve problems.M.A2HS.24Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.M.A2HS.25Represent 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.)M.A2HS.26Rearrange 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. For example, finding the equation of a line through a given point perpendicular to another line allows one to find the distance from a point to a line. This example applies to earlier instances of this standard, not to the current course.M.A2HS.27For 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.M.A2HS.28Relate 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.) Note: Emphasize the selection of a model function based on behavior of data and context.M.A2HS.29Calculate 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. Note: Emphasize the selection of a model function based on behavior of data and context.M.A2HS.3Solve quadratic equations with real coefficients that have complex solutions.M.A2HS.30Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.M.A2HS.31Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.M.A2HS.32Compare 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.)M.A2HS.33Write 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.)M.A2HS.34Identify 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.M.A2HS.35Find 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 x^3 or f(x) = (x+1)/(x-1) for x ≠ 1.)M.A2HS.36For exponential models, express as a logarithm the solution to a b^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.M.A2HS.37Use 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.M.A2HS.38Understand statistics as a process for making inferences about population parameters based on a random sample from that population.M.A2HS.39Decide 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?)M.A2HS.4Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x – 2i).M.A2HS.40Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.M.A2HS.41Use 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.M.A2HS.42Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.M.A2HS.43Evaluate reports based on data.M.A2HS.44Use probabilities to make fair decisions (e.g., drawing by lots or using a random number generator).M.A2HS.45Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, and/or pulling a hockey goalie at the end of a game).M.A2HS.5Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.M.A2HS.6Interpret expressions that represent a quantity in terms of its context.M.A2HS.7Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 – (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 – y^2)(x^2 + y^2).M.A2HS.8Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.M.A2HS.9Understand 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.M.A2HS.ICDInferences and Conclusions from DataM.A2HS.MFModeling with FunctionsM.A2HS.PRRPolynomial, Rational, and Radical RelationshipsM.A2HS.TFTrigonometric Functions
Example Problems
Simplify:
Does the polynomial have as a factor?
Factor completely:
Expand:
Expand and combine like terms:
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