Algebra II

MAL2: Algebra II

MAL2.A: Algebra

MAL2.B: Statistics and Probability

MAL2.C: Functions

MAL2.D: Number and Quantity

MAL2.A.1: solve quadratic equations in one variable

MAL2.A.2: interpret expressions that represent a quantity in terms of its context

MAL2.A.3: interpret parts of an expression such as terms, factors, and coefficients

MAL2.A.4: interpret complicated expressions by viewing one or more of their parts as a single entity (e.g., interpret P(1+r) as the product of P and a factor not depending on P)

MAL2.A.5: use the structure of an expression to rewrite it in different equivalent forms [e.g., see x4 - y4 as ((x²) -(y²))², thus recognizing it as a difference of squares that can be factored as (x²-y²)(x²+y²)]

MAL2.A.6: choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression

MAL2.A.7: use the properties of exponents to transform expressions for exponential functions

MAL2.A.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 (e.g., calculate mortgage payments)

MAL2.A.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

MAL2.A.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)

MAL2.A.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

MAL2.A.12: prove polynomial identities and use them to describe numerical relationships (e.g., the polynomial identity (x² + y²)² = (x² - y²)² + (2xy)² can be used to generate Pythagorean triples)

MAL2.A.13: know and apply that the Binomial Theorem gives 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 using Pascal’s Triangle

MAL2.A.14: rewrite simple rational expressions in different forms using inspection, long division, or a computer algebra system; 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)

MAL2.A.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

MAL2.A.16: create equations and inequalities in one variable and use them to solve problems (i.e., create equations in one variable that describes simple rational functions, exponential functions, etc.)

MAL2.A.17: create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales (i.e., create and graph equations in two variables to describe radical functions, rational functions, etc.)

MAL2.A.18: represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context (e.g., represent inequalities describing nutritional and cost constraints on combinations of different foods)

MAL2.A.19: rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations

MAL2.A.20: solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise

MAL2.A.21: 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 a graph, find the solution to a system of equations where f(x) and/or g(x) are rational functions)

MAL2.A.22: solve quadratic equations by inspection (e.g., x² = -49), taking square roots, factoring, completing the square, and using the quadratic formula, as appropriate to the initial form of the equation; recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

MAL2.B.23: use statistics appropriate to the shape of the data distribution to compare center (i.e., median, mean) and spread (i.e., interquartile, range, standard deviation) of two or more different data sets

MAL2.B.24: 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

MAL2.B.25: understand statistics as a process for making inferences about population parameters based on a random sample from that population

MAL2.B.26: decide if a specified model is consistent with results from a given data-generating process (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?)

MAL2.B.27: recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each

MAL2.B.28: 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

MAL2.B.29: use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant

MAL2.B.30: evaluate reports based on data

MAL2.C.31: using tables, graphs, equations, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities; sketch a graph showing key features, including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior

MAL2.C.32: 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)

MAL2.C.33: 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

MAL2.C.34: graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases

MAL2.C.35: graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions

MAL2.C.36: graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior

MAL2.C.37: graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior

MAL2.C.38: graph exponential and logarithmic functions, showing intercepts and end behavior

MAL2.C.39: write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function

MAL2.C.40: use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context

MAL2.C.41: use the properties of exponents to interpret expressions for exponential functions

MAL2.C.42: compare properties of two functions each represented algebraically, graphically, numerically in tables, and/or by a verbal description

MAL2.C.43: write a function that describes a relationship between two quantities (e.g., quadratic, polynomial, rational, radical, exponential, logarithmic)

MAL2.C.44: 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)

MAL2.C.45: compose functions

MAL2.C.46: identify the effect on the graph of replacing f(x) by f(x) + k, kf(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

MAL2.C.47: find inverse functions

MAL2.C.48: 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)

MAL2.C.49: verify by composition that one function is the inverse of another

MAL2.C.50: read values of an inverse function from a graph or a table, given that the function has an inverse

MAL2.C.51: understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents

MAL2.C.52: express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology

MAL2.D.53: extend polynomial identities to the complex numbers (e.g., rewrite x² + 4 as (x + 2i)(x - 2i))

MAL2.D.54: know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials

MAL2.D.55: 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

MAL2.D.56: use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers

MAL2.D.57: find the conjugate of a complex number; use conjugates to find quotients of complex numbers

MAL2.D.58: solve quadratic equations with real coefficients that have complex solutions

MAL2.D.59: explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents

MAL2.D.60: rewrite expressions involving radicals and rational exponents using the properties of exponents