# Courses of Study : Mathematics

Number and Quantity
The Complex Number System
Perform arithmetic operations with complex numbers.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
1. Define the constant ein a variety of contexts.

Example: the total interest earned if a 100% annual rate is continuously compounded.

a. Explore the behavior of the function y=ex and its applications.

b. Explore the behavior of ln(x), the logarithmic function with base e, and its applications.
Unpacked Content Evidence Of Student Attainment:
Students:
• Recognize natural exponential and natural logarithmic functions by analyzing their end behavior.
• Use natural exponential and natural logarithmic functions to model a wide variety of behaviors in the world.
Teacher Vocabulary:
• Continuous
• Explore
• Behavior
• Applications
Knowledge:
Students know:
• Exponential forms y=a-bx and y=A0ek-x.
• b must be nonnegative.
• A is the initial value.
• If b>1, the function models exponential growth.
• If 0
Skills:
Students are able to:
• Use natural exponential functions to describe the growth of natural phenomena.
• Use natural logarithm models to describe the time needed for the growth of natural phenomena.
Understanding:
Students understand that:
• ln(x) gives the time needed to grow x.
• Ex gives the amount of growth after the time x.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
2. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

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Students:
• Given a complex number, find the conjugate and the modulus.
• Given a quotient of complex numbers, multiply by 1 by multiplying the numerator and denominator by the conjugate of the denominator to write the indicated quotient as a single complex number.
Teacher Vocabulary:
• Conjugate
• Complex number
• Modulus/Moduli
Knowledge:
Students know:
• The definition of the conjugate of a complex number.
• A complex number divided by itself equals 1.
• The product of a complex number and its conjugate is a real number (the square of the modulus).
Skills:
Students are able to:
• Find the conjugate of a complex number.
• Find the modulus of a complex number
• Find the product of two complex numbers.
• Find (simplify) the quotient of complex numbers.
Understanding:
Students understand that:
• The conjugate of a complex number differs by the sign of its imaginary part and has the same modulus.
• The modulus of a complex number corresponds to the magnitude of a vector and, therefore, is useful in the geometric representation of complex numbers.
• Mathematical convention is that radical expressions are not left in denominators to facilitate numerical approximations. therefore, since the i is equal to the square root of -1, conventional form says that i does not appear in the denominator of a fraction.
• Different forms of a complex number quotient (indicated quotient, single complex number) may be more useful for various purposes.
Diverse Learning Needs:
Represent complex numbers and their operations on the complex plane.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
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.

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Students:
• Given any complex number.
• Graphically represent the number on the complex plane in rectangular and polar form.
• Justify that the rectangular (a+bi) and polar forms (z = r(cosΘ. +iSinΘ.)) are equivalent.
Teacher Vocabulary:
• Complex plane
• Polar form
Knowledge:
Students know:
• In the complex plane the horizontal axis is the real axis (a) and the vertical axis is the imaginary axis (b).
• Trigonometric techniques for finding measures of angles and coordinates on the unit circle.
• The characteristics of the polar coordinate system.
• Techniques for plotting polar coordinates.
Skills:
Students are able to:
• Use trigonometry to find the measures of angles and coordinates on the unit circle.
• Use the Pythagorean Theorem to find the lengths of sides of a right triangle.
• Convert between polar and rectangular forms.
• Plot polar coordinates.
Understanding:
Students understand that:
• A complex number (a+bi) can be graphed in a rectangular coordinate system as (a, b).
• A complex number may be represented in the plane using equivalent polar and rectangular coordinates.
• Different representations of a complex number may be more useful for various purposes.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
4. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

Example: (-1+ √3i)3=8 because (-1+ √3i) has modulus 2 and argument 120o.
Unpacked Content Evidence Of Student Attainment:
Students:
Given any two complex numbers, (a+bi and c+di),
• Find algebraically the sum, difference, and product of the two complex numbers and identify conjugates of the complex numbers.
• Represent the two vectors, their conjugates and their sum, difference, and product geometrically in the complex plane and connect these representations to the algebraic results.
• Generalize findings from geometric representations of sums, differences, products, and conjugates of complex numbers to properties (e.g., conjugate is the reflection about the x-axis, DeMoivre's Theorem) and use these properties in calculations.
Teacher Vocabulary:
• Conjugation of complex numbers
• Complex plane
• Argument
Knowledge:
Students know:
• Complex numbers are represented geometrically in the complex plane with the real part measured on the x-axis and the imaginary is represented on the y-axis.
• Complex numbers can be added or subtracted by combining the real parts and the imaginary parts or by using vector procedures geometrically (end-to-end, parallelogram rule).
• The product of complex numbers in polar form may be found by multiplying the magnitudes and adding the arguments.
Skills:
Students are able to:
• Add, subtract, and multiply the component parts of complex numbers to find sums, differences, and products.
• Identify the conjugate of a complex number and use this as a computational aid, e.g., to find a quotient of complex numbers.
• Represent complex numbers in the complex plane.
• To add and subtract complex numbers geometrically.
• Multiply complex numbers in polar form.
Understanding:
Students understand that:
• Different representations of mathematical concepts (e.g., algebraic and geometric representation of complex numbers) reveal different features of the concept and each may facilitate computation and sense making in different settings.
• Mathematics is a coherent whole and structure within mathematics allows for procedures from one area to be used in another (e.g., coordinate geometry and the complex plane, vectors and complex numbers, or plotting of a conjugate of a complex number in transformational geometry).
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
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.

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Use complex numbers in polynomial identities and equations.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
6. Analyze possible zeros for a polynomial function over the complex numbers by applying the Fundamental Theorem of Algebra, using a graph of the function, or factoring with algebraic identities.

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Limits
Understand limits of functions.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
7. Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.

a. Apply limits of functions at specific values and at infinity in problems involving convergence and divergence.

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Vector and Matrix Quantities
Represent and model with vector quantities.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
8. Explain that vector quantities have both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.

Examples: v, |v|, ||v||, v.
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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
9. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
10. Solve problems involving velocity and other quantities that can be represented by vectors.

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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
11. Find the scalar (dot) product of two vectors as the sum of the products of corresponding components and explain its relationship to the cosine of the angle formed by two vectors.

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Perform operations on vectors.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0

a. Add vectors end-to-end, component-wise, and by the parallelogram rule, understanding that the magnitude of a sum of two vectors is not always the sum of the magnitudes.

b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

c. Explain 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.
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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
13. Multiply a vector by a scalar.

a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise.

Example: c(vx, vy) = (cvx, cvy)

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).
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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
14. 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.

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Algebra
Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
15. 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, extending to infinite geometric series.

Examples: calculate mortgage payments; determine the long-term level of medication if a patient takes 50 mg of a medication every 4 hours, while 70% of the medication is filtered out of the patient's blood.
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Arithmetic With Polynomials and Rational Expressions
Understand the relationship between zeros and factors of polynomials.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
16. Derive 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).
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Use polynomial identities to solve problems.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
17. 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.
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Rewrite rational expressions.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
18. 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), using inspection, long division, or, for the more complicated cases, a computer algebra system.
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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
19. Add, subtract, multiply, and divide rational expressions.

a. Explain why rational expressions form a system analogous to the rational numbers, which is closed under addition, subtraction, multiplication, and division by a non-zero rational expression.

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Reasoning With Equations and Inequalities
Understand solving equations as a process of reasoning and explain the reasoning.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
20. Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a clear-cut solution. Construct a viable argument to justify a solution method. Include equations that may involve linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, and trigonometric functions, and their inverses.
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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
21. Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.

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Solve systems of equations.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
22. Represent a system of linear equations as a single matrix equation in a vector variable.

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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
23. 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 x 3 or greater).

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Functions
Interpreting Functions
Interpret functions that arise in applications in terms of the context.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.

Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.
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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
25. 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. Extend from polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions.

a. Find the difference quotient f(x+�"x)-f(x)/�"x of a function and use it to evaluate the average rate of change at a point.

b. Explore how the average rate of change of a function over an interval (presented symbolically or as a table) can be used to approximate the instantaneous rate of change at a point as the interval decreases.
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Analyze functions using different representations.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
26. Graph functions expressed symbolically and show key features of the graph, by hand and using technology. Use the equation of functions to identify key features in order to generate a graph.

a. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

b. Graph trigonometric functions and their inverses, showing period, midline, amplitude, and phase shift.
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Building Functions
Build a function that models a relationship between two quantities.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
27. Compose functions. Extend to polynomial, trigonometric, radical, and rational functions.

Example: 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.
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Build new functions from existing functions.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
28. Find inverse functions.

a. Given that a function has an inverse, write an expression for the inverse of the function.

Example: Given f(x) = 2x3 or f(x) = (x + 1)/(x - 1) for x ≠ 1 find f-1(x).

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

c. Read values of an inverse function from a graph or a table, given that the function has an inverse.

d. Produce an invertible function from a non-invertible function by restricting the domain.
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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
29. Use the inverse relationship between exponents and logarithms to solve problems involving logarithms and exponents. Extend from logarithms with base 2 and 10 to a base of e.
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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
30. Identify the effect on the graph of replacing f(x) by f(x)+k, k · f(x), f(k · x), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Extend the analysis to include all trigonometric, rational, and general piecewise-defined functions with and without technology.

Example: Describe the sequence of transformations that will relate y=sin(x) and y=2sin(3x).
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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
31. Graph conic sections from second-degree equations, extending from circles and parabolas to ellipses and hyperbolas, using technology to discover patterns.

a. Graph conic sections given their standard form.

Example: The graph of x2/9 + (y-3)2/4=1 will be an ellipse centered at (0,3) with major axis 3 and minor axis 2, while the graph of x2/9 + (y-3)2/4=1 will be a hyperbola centered at (0,3) with asymptotes with slope ±3/2.

b. Identify the conic section that will be formed, given its equation in general form.

Example: 5y2 - 25x2=-25 will be a hyperbola.
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Trigonometric Functions
Recognize attributes of trigonometric functions and solve problems involving trigonometry.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
32. Solve application-based problems involving parametric and polar equations.

a. Graph parametric and polar equations.

b. Convert parametric and polar equations to rectangular form.
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Extend the domain of trigonometric functions using the unit circle.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
33. Use special triangles to determine geometrically the values of sine, cosine, and 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.
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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
34. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

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Model periodic phenomena with trigonometric functions.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
35. Demonstrate that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

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 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
36. 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.

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Prove and apply trigonometric identities.
 Mathematics (2019) Grade(s): 9 - 12 Precalculus All Resources: 0
37. Use trigonometric identities to solve problems.

a. Use the Pythagorean identity sin2 (θ) + cos2(θ) = 1 to derive the other forms of the identity.

Example: 1 + cot2 (θ) = csc2 (θ)

b. Use the angle sum formulas for sine, cosine, and tangent to derive the double angle formulas.

c. Use the Pythagorean and double angle identities to prove other simple identities.
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