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### NQ Number and Quantity

- NQ.1.PC Students will use complex numbers and determine how polar and rectangular coordinates are related
- NQ.1.PC.1 Find the conjugate of a complex number. Use conjugates to find quotients of complex numbers. Use conjugates to find moduli.
- NQ.1.PC.2 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers). Explain why the rectangular and polar forms of a given complex number represent the same number.
- NQ.1.PC.3 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; Use properties of geometrical representation for computation.
- NQ.1.PC.4 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.

- NQ.2.PC Students will perform operations with vectors and use those skills to solve problems
- NQ.2.PC.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*). - NQ.2.PC.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
- NQ.2.PC.3 Solve problems involving velocity and other quantities that can be represented by vectors.
- NQ.2.PC.4 Add and subtract vectors. 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. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. 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. Perform vector subtraction component-wise. - NQ.2.PC.5 Multiply a vector by a scalar. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; Perform scalar multiplication component-wise, e.g., as
*c*(*v*_{x},*v*subscript*y*) = (*c**v*_{x},*c**v*subscript*y*). Compute the magnitude of a scalar multiple*c***v**using ||*c***v**|| = |*c*|**v**. Compute the direction of*c***v**knowing that when |*c*|**v**≠ 0, the direction of*c***v**is either along**v**(for*c*> 0) or against**v**(for*c*< 0). - NQ.2.PC.6 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.

- NQ.2.PC.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.,

### T Trigonometry

- T.3.PC Students will develop and apply the definitions of the six trigonometric functions and use the definitions to solve problems and verify identities
- T.3.PC.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
- T.3.PC.2 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 around the unit circle.
- T.3.PC.3 Use special right triangles to determine geometrically the exact values of sine, cosine, tangent for π/3, π/4, π/6, and π/2. 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 exact values for*x*, where*x*is any real number. - T.3.PC.4 Develop the Pythagorean identity, sin²(θ) + cos²(θ) = 1. Given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle, use the Pythagorean identity to find the remaining trigonometric functions.
- T.3.PC.5 Develop the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
- T.3.PC.6 Derive the formula
*A*= 1/2*a**b*sin(*C*) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. - T.3.PC.7 Prove the Law of Sines and the Law of Cosines and use them to solve problems.
- T.3.PC.8 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.
- T.3.PC.9 Define and use reciprocal functions, cosecant, secant, and cotangent to solve problems.

- T.4.PC Students will solve trigonometric equations and sketch the graph of periodic trigonometric functions
- T.4.PC.1 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
- T.4.PC.2 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
- T.4.PC.3 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
- T.4.PC.4 Use inverse functions to: Solve trigonometric equations that arise in modeling context(s); evaluate the solutions of trigonometric equations, with or without technology, and interpret the solutions of trigonometric equations in terms of the context(s).
- T.4.PC.5 Recognize that some trigonometric equations have infinitely many solutions and be able to state a general formula to represent the infinite solutions.

### CS Conic Sections

- CS.5.PC Students will identify, analyze, and sketch the graphs of the conic sections and relate their equations and graphs
- CS.5.PC.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem. Complete the square to find the center and radius of a circle given by an equation.
- CS.5.PC.2 Derive the equation of a parabola given a focus and directrix.
- CS.5.PC.3 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.
- CS.5.PC.4 Find the equations for the asymptotes of a hyperbola.
- CS.5.PC.5 Complete the square in order to generate an equivalent form of an equation for a conic section; use that equivalent form to identify key characteristics of the conic section.
- CS.5.PC.6 Identify, graph, write, and analyze equations of each type of conic section, using properties such as symmetry, intercepts, foci, asymptotes, and eccentricity, and using technology when appropriate.
- CS.5.PC.7 Solve systems of equations and inequalities involving conics and other types of equations, with and without appropriate technology.

### F Functions

- F.6.PC Students will be able to find the inverse of functions and use composition of functions to prove that two functions are inverses
- F.6.PC.1 Write a function that describes a relationship between two quantities. From a context, determine an explicit expression, a recursive process, or steps for calculation. Combine standard function types using arithmetic operations. (e.g., given that
*f*(*x*) and*g*(*x*) are functions developed from a context, find (f + g)(x), (f – g)(x), (fg)(x), (f/g)(x), and any combination thereof, given*g*(x) ≠ 0.) Compose functions. - F.6.PC.2 Find inverse functions. Solve an equation of the form
*y*=*f*(*x*) for a simple function*f*that has an inverse and write an expression for the inverse. For example,*f*(*x*) = 2 x² or (x) = (x + 1)/(x– 1) for*x*≠ 1. Verify by composition that one function is the inverse of another. Read values of an inverse function from a graph or a table, given that the function has an inverse. Produce an invertible function from a non-invertible function by restricting the domain. - F.6.PC.3 Understand the inverse relationship between exponents and logarithms. Use the inverse relationship between exponents and logarithms to solve problems.

- F.6.PC.1 Write a function that describes a relationship between two quantities. From a context, determine an explicit expression, a recursive process, or steps for calculation. Combine standard function types using arithmetic operations. (e.g., given that
- F.7.PC Students will be able to interpret different types of functions and their key characteristics including polynomial, exponential, logarithmic, power, trigonometric, rational, and other types of functions
- F.7.PC.1 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
- F.7.PC.2 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.
- F.7.PC.3 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. - F.7.PC.4 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.
- F.7.PC.5 Calculate and interpret the average rate of change of a function (presented algebraically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F.7.PC.6 Graph functions expressed algebraically and show key features of the graph, with and without technology. Graph linear and quadratic functions and, when applicable, show intercepts, maxima, and minima. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph power and polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph exponential and logarithmic functions, showing intercepts and end behavior. Graph trigonometric functions, showing period, midline, and amplitude.
- F.7.PC.7 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
- F.7.PC.8 Build functions to model real-world applications using algebraic operations on functions and composition, with and without appropriate technology. (e.g., profit functions as well as volume and surface area, optimization subject to constraints).