NY Math Standards (Algebra II)

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.

Use the Binomial Theorem for the expansion of (x + y)^n for a positive integer n.

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.

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).

Identify zeros of polynomial functions when suitable factorizations are available. (Shared standard with Algebra I)

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). Note: This standard is a fluency expectation for Algebra II. This standard sets an expectation that students will divide polynomials with remainders by inspection in simple cases. For example, one can view the rational expression (x+4)/(x+3) as ((x+3)+1)/(x+3) which is 1 + 1/(x+3).

Create equations and inequalities in one variable to represent a real-world context. (Shared standard with Algebra I) Note: This is strictly the development of the model (equation/inequality). Tasks include linear, quadratic, rational, and exponential functions.

Solve systems of linear equations in three variables.

Represent a system of linear equations as a single matrix equation in a vector variable.

Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3×3 or greater).

Explain each step when solving rational or radical equations as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Solve rational and radical equations in one variable, identify extraneous solutions, and explain how they arise. Note: Radical equations may include but are not limited to those of the form x^(3/5) = 8 and 3x^(3/4) + 5 = 86.

Solve a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. (Shared standard with Algebra I) Note: Conics are limited to parabolas and circles.

Given the equations y = f(x) and y = g(x): recognize that each x-coordinate of the intersection(s) is the solution to the equation f(x) = g(x); find the solutions approximately using technology to graph the functions or make tables of values; find the solution of f(x) < g(x) or f(x) ≤ g(x) graphically; and interpret the solution in context. ★ (Shared standard with Algebra I) Note: Tasks include cases where f(x) and/or g(x) are linear, polynomial, absolute value, square root, cube root, trigonometric, exponential, and logarithmic functions.

Recognize and use the structure of an expression to identify ways to rewrite it. (Shared standard with Algebra I) e.g., 81x⁴ - 16y⁴ is equivalent to (9x²)² - (4y²)² or (9x² - 4y²)(9x² + 4y²) or (3x + 2y)(3x - 2y)(9x² + 4y²); (x²+4)/(x²+3) is equivalent to ((x²+3)+1)/(x²+3) = 1 + 1/(x²+3); 3x³ - 5x² - 48x + 80 is equivalent to 3x(x²-16) - 5(x²-16), which when factored completely is (3x-5)(x+4)(x-4). Notes: Includes factoring by grouping and factoring the sum and difference of cubes. Tasks are limited to polynomial, rational, or exponential expressions. Quadratic expressions include leading coefficients other than 1. This standard is a fluency expectation for Algebra II.

Compose functions and state resulting domain. ★ e.g., if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. Note: The domain of a resulting composition function could differ from the domains of the individual functions.

Determine algebraically whether or not a function is even or odd.

Verify by composition that one function is the inverse of another.

Given the graph or table of an invertible function, determine coordinates of its inverse.

Determine an invertible function from a non-invertible function by restricting the domain. e.g., inverse trigonometric functions.

Use inverse relationships to solve problems involving logarithms and exponents.

Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms and solve logarithmic equations.

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Note: In Algebra II, sequences will be defined/written recursively and explicitly in subscript notation.

Using f(x) + k, k f(x), f(kx), and f(x + k): identify the effect on the graph when 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; write a new function using the value of k; and use technology to experiment with cases and explore the effects on the graph. Include recognizing even and odd functions from their graphs. (Shared standard with Algebra I) Note: Algebra II tasks may involve polynomial, square root, cube root, exponential, logarithmic, and trigonometric functions.

Find the inverse of a one-to-one function both algebraically and graphically.

Understand inverse relationships between exponents and logarithms algebraically and graphically.

Represent and evaluate the sum of a finite arithmetic or finite geometric series, using summation (sigma) notation.

Explore the derivation of the formulas for finite arithmetic and finite geometric series. Use the formulas to solve problems. ★

Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available. ★

Recognize that a sequence is a function whose domain is a subset of the integers. (Shared standard with Algebra I) Notes: In Algebra II, sequences will be defined/written recursively and explicitly in subscript notation. This standard is a fluency expectation for Algebra II. Fluency in translating between recursive definitions and closed forms is helpful when dealing with many problems involving sequences and series, with applications ranging from fitting functions to tables to problems in finance.

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. (Shared standard with Algebra I) Notes: Algebra II key features include: intercepts, zeros; intervals where the function is increasing, decreasing, positive, or negative; relative maxima and minima; symmetries; end behavior; and periodicity. Tasks may involve real-world context and may include polynomial, square root, cube root, exponential, logarithmic, and trigonometric functions.

Calculate and interpret the average rate of change of a function over a specified interval. (Shared standard with Algebra I) Notes: Functions may be presented by function notation, a table of values, or graphically. Algebra II tasks have a real-world context and may involve polynomial, square root, cube root, exponential, logarithmic, and trigonometric functions.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (Shared standard with Algebra I) Note: Tasks may involve polynomial, square root, cube root, exponential, logarithmic, and trigonometric functions.

Construct a linear or exponential function symbolically given: a graph; a description of the relationship; and two input-output pairs (include reading these from a table). (Shared standard with Algebra I)

Use logarithms to solve exponential equations, such as ab^(ct) = d (where a, b, c, and d are real numbers and b > 0) and evaluate the logarithm using technology.

Interpret the parameters in a linear or exponential function in terms of a context. (Shared standard with Algebra I) Note: Algebra II tasks have a real-world context and exponential functions are not limited to integer domains.

Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number.

Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Solve trigonometric equations: analytically with inverse functions and graphically with technology, and interpret solutions in terms of the context. ★

Prove the sum and difference formulas for sine, cosine, and tangent and use them to solve problems.

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

Apply concepts of the unit circle in the coordinate plane to calculate the values of the six trigonometric functions given angles in radian measure.

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Note: Focus of this standard is on cos(x), sin(x) and tan(x).

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, horizontal shift, and midline.

Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1. Find the value of any of the six trigonometric functions given any other trigonometric function value and when necessary find the quadrant of the angle.

Construct a tangent line from a point outside a given circle to the circle.

Explore the relationship among the parabola, focus, and directrix and use the equation to model a real-life situation, using technology as appropriate. ★ Note: Explore indicates that the topic is an important concept that builds the foundation for progression toward mastery in later grades.

Derive the equations of ellipses and hyperbolas given the foci.

Use these equations to model real life situations. ★

Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

Prove the Law of Sines and the Law of Cosines and apply in all cases, including the ambiguous case.

Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. e.g., surveying problems or problems that involve resultant forces.

Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and convert between rectangular and polar forms of a given complex number.

Determine whether rectangular or polar form is more efficient given the context.

Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. e.g., (-1 + √3 i)^3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.

Calculate the distance between two points in the complex plane. Note: Standard extends the distance and midpoint calculation from the Cartesian coordinate plane to the complex plane.

Find the midpoint of the segment whose endpoints are in the complex plane.

Extend polynomial identities to the complex numbers. e.g., Rewrite x² + 4 as (x + 2i)(x - 2i).

State the Fundamental Theorem of Algebra and use it to find roots of polynomials.

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.

Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Note: Tasks include simplifying powers of i.

Explore how the meaning of rational exponents follows from extending the properties of integer exponents. e.g., We define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^3 = 5^((1/3)·3) to hold, so (5^(1/3))^3 must equal 5.

Convert between radical expressions and expressions with rational exponents using the properties of exponents. Note: All radical expressions involving variables assume the variables are representing positive numbers. Includes expressions with variable factors, such as the cube root of 27x^5y^3, being equivalent to (27x^5y^3)^(1/3) which equals 3x(x^(2/3))y.

Represent a vector analytically and geometrically. e.g., rectangular form, polar form, unit form.

Find the magnitude and direction of a given vector.

Solve problems using vectors analytically and geometrically. e.g., velocity and forces.

Add and subtract vectors analytically and geometrically.

Multiply a vector by a scalar analytically and geometrically.

Use matrices to represent and model real world situations. e.g., networks.

Multiply matrices by scalars.

Add, subtract, and multiply matrices.

Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

Use matrices to perform linear transformations in the plane. e.g., multiplying a vector by a 2×2 matrix.

Calculate and interpret the determinant of a matrix. e.g., calculating area.

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

Interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and calculate conditional probabilities.

Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.

Solve problems using permutations and combinations to compute probabilities of compound events.

Informally assess the fit of a function by plotting and analyzing residuals.

Recognize whether or not a normal curve is appropriate for a given data set.

If appropriate, determine population percentages using a graphing calculator for an appropriate normal curve.

Determine if a value for a sample proportion or sample mean is likely to occur based on a given simulation. Note: For the purposes of this course, if the statistic falls within two standard deviations of the mean (95% interval centered on the population parameter), then the statistic is considered likely (plausible, usual).

Recognize the purposes of and differences among surveys, experiments, and observational studies. Explain how randomization relates to each.

Given a simulation model based on a sample proportion or mean, construct the 95% interval centered on the statistic (+/- two standard deviations) and determine if a suggested parameter is plausible.

Use the tools of statistics to draw conclusions from numerical summaries.

Use the language of statistics to critique claims from informational texts. e.g., causation vs correlation, bias, measures of center and spread.

Define a random variable for a quantity of interest.

Graph a probability distribution for a discrete random variable based on either empirical or theoretical probabilities.

Calculate and interpret the expected value of a random variable. e.g., Find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. e.g., Find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

Use expected values from probability distributions to evaluate and compare the outcomes of decisions. e.g., Compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

Use probabilities to make fair decisions. e.g., Determine if a decision-making strategy produces equally probable outcomes.

Using probability concepts, evaluate decisions and strategies. e.g., Make decisions based on the most favorable outcome.