HS.N-CN.A. The Complex Number System: Perform arithmetic operations with complex numbers.

• Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. (CCSS: HS.N-CN.A.2)

HS.A-APR.B. Arithmetic with Polynomials & Rational Expressions: Understand the relationship between zeros and factors of polynomials.

• 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). (Students need not apply the Remainder Theorem to polynomials of degree greater than 4.) (CCSS: HS.A-APR.B.2)

• 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. (CCSS: HS.A-APR.B.3)

HS.F-IF.B. Interpreting Functions: Interpret functions that arise in applications in terms of the context.

• 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (CCSS: HS.F-IF.B.4)

HS.F-IF.C. Interpreting Functions: Analyze functions using different representations.

• Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.⭑ (CCSS: HS.F-IF.C.7)

  • Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. (CCSS: HS.F-IF.C.7.c)

HS.F-BF.B. Building Functions: Build new functions from existing functions. 

• 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. (CCSS: HS.F-BF.B.3)

A star symbol (⭑) represents grade level expectations and evidence outcomes that make up a mathematical modeling standards category.

HS.N-CN.A. The Complex Number System: Perform arithmetic operations with complex numbers.

• (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. (CCSS: HS.N-CN.A.3)

HS.N-CN.C. The Complex Number System: Use complex numbers in polynomial identities and equations.

• (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i). (CCSS: HS.N-CN.C.8)

• (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. (CCSS: HS.N-CN.C.9)

HS.A-APR.C. Arithmetic with Polynomials & Rational Expressions: Use polynomial identities to solve problems.

• (+) Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 − y2)2 + (2xy)2 can be used to generate Pythagorean triples. (CCSS: HS.A-APR.C.4)

• (+) 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. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) (CCSS: HS.A-APR.C.5)

HS.A-APR.D. Arithmetic with Polynomials & Rational Expressions: Rewrite rational expressions.

• (+) 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. (CCSS: HS.A-APR.D.7)

HS.F-IF.C. Interpreting Functions: Analyze functions using different representations.

• Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.⭑ (CCSS: HS.F-IF.C.7)

  • (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. (CCSS: HS.F-IF.C.7.d)

HS.F-BF.A. Building Functions: Build a function that models a relationship between two quantities.

• Write a function that describes a relationship between two quantities.⭑ (CCSS: HS.F-BF.A.1)

  • (+) Compose functions. For 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. (CCSS: HS.F- BF.A.1.c)

HS.F-BF.B. Building Functions: Build new functions from existing functions. 

• Find inverse functions. (CCSS: HS.F-BF.B.4)

  • (+) Verify by composition that one function is the inverse of another. (CCSS: HS.F-BF.B.4.b)

  • (+) Read values of an inverse function from a graph or table, given that the function has an inverse. (CCSS: HS.F-BF.B.4.c)

  • (+) Produce an invertible function from a non-invertible function by restricting the domain. (CCSS: HS.F-BF.B.4.d)

• (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. (CCSS: HS.F-BF.B.5)

HS.S-CP.B. Conditional Probability & the Rules of Probability: Use the rules of probability to compute probabilities of compound events in a uniform probability model.

• (+) Use permutations and combinations to solve problems. (CCSS: HS.S-CP.B.9) (Binomial Theorem)

HS.G-GPE.A. Expressing Geometric Properties with Equations: Translate between the geometric description and the equation for a conic section.

• (+) Derive the equation of a parabola given a focus and directrix. (CCSS: HS.G-GPE.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. (CCSS: HS.G-GPE.A.3)

A star symbol (⭑) represents grade level expectations and evidence outcomes that make up a mathematical modeling standards category.

Please visit the complete 2020 Colorado Academic Standards for High School Mathematics to view the following:

• Colorado Essential Skills and Mathematical Practices connections

• Inquiry Questions

• Coherence Connections

HS.A-SSE.B. Seeing Structure in Expressions: Write expressions in equivalent forms to solve problems.

• Use the formula for the sum of a finite geometric series (when the common ratio is not 1) to solve problems. For example, calculate mortgage payments.⭑ (CCSS: HS.A-SSE.B.4)

  • (+) Derive the formula for the sum of a finite geometric series (when the common ratio is not 1). (CCSS: HS.A-SSE.B.4)

HS.F-BF.A. Building Functions: Build a function that models a relationship between two quantities. 

• Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.⭑ (CCSS: HS.F-BF.A.2) (Reinforce from Algebra 1)