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### P.N Number and Quantity

- P.N.NE Number Expressions
- P.N.NE.A Represent, interpret, compare, and simplify number expressions.
- P.N.NE.A.1 Use the laws of exponents and logarithms to expand or collect terms in expressions; simplify expressions or modify them in order to analyze them or compare them.
- P.N.NE.A.2 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
- P.N.NE.A.3 Classify real numbers and order real numbers that include transcendental expressions, including roots and fractions of π and e.
- P.N.NE.A.4 Simplify complex radical and rational expressions; discuss and display understanding that rational numbers are dense in the real numbers and the integers are not.
- P.N.NE.A.5 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

- P.N.NE.A Represent, interpret, compare, and simplify number expressions.
- P.N.CN The Complex Number System
- P.N.CN.A Perform complex number arithmetic and understand the representation on the complex plane.
- P.N.CN.A.1 Perform arithmetic operations with complex numbers expressing answers in the form a + bi.
- P.N.CN.A.2 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
- P.N.CN.A.3 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
- P.N.CN.A.4 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
- P.N.CN.A.5 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

- P.N.CN.B Use complex numbers in polynomial identities and equations.
- P.N.CN.B.6 Extend polynomial identities to the complex numbers.
- P.N.CN.B.7 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

- P.N.CN.A Perform complex number arithmetic and understand the representation on the complex plane.
- P.N.VM Vector and Matrix Quantities
- P.N.VM.A Represent and model with vector quantities.
- P.N.VM.A.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
- P.N.VM.A.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
- P.N.VM.A.3 Solve problems involving velocity and other quantities that can be represented by vectors.

- P.N.VM.B Understand the graphic representation of vectors and vector arithmetic.
- P.N.VM.B.4 Add and subtract vectors.
- P.N.VM.B.4.a Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
- P.N.VM.B.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
- P.N.VM.B.4.c Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

- P.N.VM.B.5 Multiply a vector by a scalar.
- P.N.VM.B.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v
_{x}, v_{y}) = (cv_{x}, cv_{y}). - P.N.VM.B.5.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

- P.N.VM.B.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v
- P.N.VM.B.6 Calculate and interpret the dot product of two vectors.

- P.N.VM.B.4 Add and subtract vectors.
- P.N.VM.C Perform operations on matrices and use matrices in applications.
- P.N.VM.C.7 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
- P.N.VM.C.8 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
- P.N.VM.C.9 Add, subtract, and multiply matrices of appropriate dimensions.
- P.N.VM.C.10 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
- P.N.VM.C.11 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
- P.N.VM.C.12 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
- P.N.VM.C.13 Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

- P.N.VM.A Represent and model with vector quantities.

### P.A Algebra

- P.A.S Sequences and Series
- P.A.S.A Understand and use sequences and series.
- P.A.S.A.1 Demonstrate an understanding of sequences by representing them recursively and explicitly.
- P.A.S.A.2 Use sigma notation to represent a series; expand and collect expressions in both finite and infinite settings.
- P.A.S.A.3 Derive and use the formulas for the general term and summation of finite or infinite arithmetic and geometric series, if they exist.
- P.A.S.A.3.a Determine whether a given arithmetic or geometric series converges or diverges.
- P.A.S.A.3.b Find the sum of a given geometric series (both infinite and finite).
- P.A.S.A.3.c Find the sum of a finite arithmetic series.

- P.A.S.A.4 Understand that series represent the approximation of a number when truncated; estimate truncation error in specific examples.
- P.A.S.A.5 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.

- P.A.S.A Understand and use sequences and series.
- P.A.REI Reasoning with Equations and Inequalities
- P.A.REI.A Solve systems of equations and nonlinear inequalities.
- P.A.REI.A.1 Represent a system of linear equations as a single matrix equation in a vector variable.
- P.A.REI.A.2 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).
- P.A.REI.A.3 Solve nonlinear inequalities (quadratic, trigonometric, conic, exponential, logarithmic, and rational) by graphing (solutions in interval notation if one-variable), by hand and with appropriate technology.
- P.A.REI.A.4 Solve systems of nonlinear inequalities by graphing.

- P.A.REI.A Solve systems of equations and nonlinear inequalities.
- P.A.PE Parametric Equations
- P.A.PE.A Describe and use parametric equations.
- P.A.PE.A.1 Graph curves parametrically (by hand and with appropriate technology).
- P.A.PE.A.2 Eliminate parameters by rewriting parametric equations as a single equation.

- P.A.PE.A Describe and use parametric equations.
- P.A.C Conic Sections
- P.A.C.A Understand the properties of conic sections and model real-world phenomena.
- P.A.C.A.1 Display all of the conic sections as portions of a cone.
- P.A.C.A.2 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
- P.A.C.A.3 From an equation in standard form, graph the appropriate conic section: ellipses, hyperbolas, circles, and parabolas. Demonstrate an understanding of the relationship between their standard algebraic form and the graphical characteristics.
- P.A.C.A.4 Transform equations of conic sections to convert between general and standard form.

- P.A.C.A Understand the properties of conic sections and model real-world phenomena.

### P.F Functions

- P.F.BF Building Functions
- P.F.BF.A Build new functions from existing functions.
- P.F.BF.A.1 Understand how the algebraic properties of an equation transform the geometric properties of its graph.
- P.F.BF.A.2 Develop an understanding of functions as elements that can be operated upon to get new functions: addition, subtraction, multiplication, division, and composition of functions.
- P.F.BF.A.3 Compose functions.
- P.F.BF.A.4 Construct the difference quotient for a given function and simplify the resulting expression.
- P.F.BF.A.5 Find inverse functions (including exponential, logarithmic, and trigonometric).
- P.F.BF.A.5.a Calculate the inverse of a function, f (x) , with respect to each of the functional operations; in other words, the additive inverse, – f (x) , the multiplicative inverse, 1 / f(x), and the inverse with respect to composition, f
^{–1}(x). Understand the algebraic and graphical implications of each type. - P.F.BF.A.5.b Verify by composition that one function is the inverse of another.
- P.F.BF.A.5.c Read values of an inverse function from a graph or a table, given that the function has an inverse.
- P.F.BF.A.5.d Recognize a function is invertible if and only if it is one-to-one. Produce an invertible function from a non-invertible function by restricting the domain.

- P.F.BF.A.5.a Calculate the inverse of a function, f (x) , with respect to each of the functional operations; in other words, the additive inverse, – f (x) , the multiplicative inverse, 1 / f(x), and the inverse with respect to composition, f
- P.F.BF.A.6 Explain why the graph of a function and its inverse are reflections of one another over the line y = x.

- P.F.BF.A Build new functions from existing functions.
- P.F.IF Interpreting Functions
- P.F.IF.A Analyze functions using different representations.
- P.F.IF.A.1 Determine whether a function is even, odd, or neither.
- P.F.IF.A.2 Analyze qualities of exponential, polynomial, logarithmic, trigonometric, and rational functions and solve real-world problems that can be modeled with these functions (by hand and with appropriate technology).
- P.F.IF.A.4 Identify the real zeros of a function and explain the relationship between the real zeros and the x-intercepts of the graph of a function (exponential, polynomial, logarithmic, trigonometric, and rational).
- P.F.IF.A.5 Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax² + c.
- P.F.IF.A.6 Visually locate critical points on the graphs of functions and determine if each critical point is a minimum, a maximum, or point of inflection. Describe intervals where the function is increasing or decreasing and where different types of concavity occur.
- P.F.IF.A.7 Graph rational functions, identifying zeros, asymptotes (including slant), and holes (when suitable factorizations are available) and showing end-behavior.
- P.F.IF.A.8 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

- P.F.IF.A Analyze functions using different representations.
- P.F.TF Trigonometric Functions
- P.F.TF.A Extend the domain of trigonometric functions using the unit circle.
- P.F.TF.A.1 Convert from radians to degrees and from degrees to radians.
- P.F.TF.A.2 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 π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.
- P.F.TF.A.3 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
- P.F.TF.A.4 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

- P.F.TF.A Extend the domain of trigonometric functions using the unit circle.
- P.F.GT Graphing Trigonometric Functions
- P.F.GT.A Model periodic phenomena with trigonometric functions.
- P.F.GT.A.1 Interpret transformations of trigonometric functions.
- P.F.GT.A.2 Determine the difference made by choice of units for angle measurement when graphing a trigonometric function.
- P.F.GT.A.3 Graph the six trigonometric functions and identify characteristics such as period, amplitude, phase shift, and asymptotes.
- P.F.GT.A.4 Find values of inverse trigonometric expressions (including compositions), applying appropriate domain and range restrictions.
- P.F.GT.A.5 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
- P.F.GT.A.6 Determine the appropriate domain and corresponding range for each of the inverse trigonometric functions.
- P.F.GT.A.7 Graph the inverse trigonometric functions and identify their key characteristics.
- P.F.GT.A.8 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

- P.F.GT.A Model periodic phenomena with trigonometric functions.

### P.G Geometry

- P.G.AT Applied Trigonometry
- P.G.AT.A Use trigonometry to solve problems.
- P.G.AT.A.1 Use the definitions of the six trigonometric ratios as ratios of sides in a right triangle to solve problems about lengths of sides and measures of angles.
- P.G.AT.A.2 Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
- P.G.AT.A.3 Derive and apply the formulas for the area of sector of a circle.
- P.G.AT.A.4 Calculate the arc length of a circle subtended by a central angle.

- P.G.AT.A.5 Prove the Laws of Sines and Cosines and use them to solve problems.
- P.G.AT.A.6 Understand and apply the Law of Sines (including the ambiguous case) and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

- P.G.AT.A Use trigonometry to solve problems.
- P.G.TI Trigonometric Identities
- P.G.TI.A Apply trigonometric identities to rewrite expressions and solve equations.
- P.G.TI.A.1 Apply trigonometric identities to verify identities and solve equations. Identities include: Pythagorean, reciprocal, quotient, sum/difference, double-angle, and half-angle.
- P.G.TI.A.2 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

- P.G.TI.A Apply trigonometric identities to rewrite expressions and solve equations.
- P.G.PC Polar Coordinates
- P.G.PC.A Use polar coordinates.
- P.G.PC.A.1 Graph functions in polar coordinates.
- P.G.PC.A.2 Convert between rectangular and polar coordinates.
- P.G.PC.A.3 Represent situations and solve problems involving polar coordinates.

- P.G.PC.A Use polar coordinates.

### P.S Statistics and Probability

- P.S.MD Model with Data
- P.S.MD.A Model data using regressions equations.
- P.S.MD.A.1 Create scatter plots, analyze patterns, and describe relationships for bivariate data (linear, polynomial, trigonometric, or exponential) to model real-world phenomena and to make predictions.
- P.S.MD.A.2 Determine a regression equation to model a set of bivariate data. Justify why this equation best fits the data.
- P.S.MD.A.3 Use a regression equation, modeling bivariate data, to make predictions. Identify possible considerations regarding the accuracy of predictions when interpolating or extrapolating.

- P.S.MD.A Model data using regressions equations.