I. Perform arithmetic operations with complex numbers.
1. Know there is a complex number i such that
( i)^ 2 = −1, and every complex number has the form a + bi with a and b real.
2. Use the relation( i)^2 = −1 and the commutative, associative, and distributive properties to add, subtract, and multiply
complex numbers.
II. Use complex numbers in polynomial identities and equations. [Polynomials with real coefficients]
7. Solve quadratic equations with real coefficients that have complex solutions.
8. (+) Extend polynomial identities to the complex numbers. For example, rewrite
(x)^2 + 4 as (x + 2i)(x – 2i).
9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
I. Interpret the structure of expressions. [Polynomial and rational]
1. Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret
P(1 + r)n as the product of P and a factor not depending on P.
2. Use the structure of an expression to identify ways to rewrite it.
II. Write expressions in equivalent forms to solve problems.
4. 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. For example, calculate mortgage payments.
I. Perform arithmetic operations on polynomials. [Beyond quadratic]
1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of
addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
II. Understand the relationship between zeros and factors of polynomials.
2. 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).
3. 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.
III. Use polynomial identities to solve problems.
4. 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.
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.
IV. Rewrite rational expressions. [Linear and quadratic denominators]
6. 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 examples, a computer algebra system.
7. (+) 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.
I. Create equations that describe numbers or relationships. [Equations using all available types of expressions, including simple
root functions]
1. Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.
Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate
axes with labels and scales.
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions
as viable or non-viable options in a modeling context.
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
I) Understand solving equations as a process of reasoning and explain the reasoning. [Simple radical and rational]
2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
II) Solve equations and inequalities in one variable.
3.1 Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in
context.
III) Represent and solve equations and inequalities graphically. [Combine polynomial, rational, radical, absolute value, and
exponential functions.]
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions
of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of
values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute
value, exponential, and logarithmic functions.
I) Build a function that models a relationship between two quantities. [Include all types of functions studied.]
1. Write a function that describes a relationship between two quantities.
b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature
of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
II) Build new functions from existing functions. [Include simple radical, rational, and exponential functions; emphasize common
effect of each transformation across function types.]
3. 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.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
For example, f(x) =2x^3 or f(x) = (x + 1)/(x − 1) for x ≠ 1.
Construct and compare linear, quadratic, and exponential models and solve problems.
4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2,
10, or e; evaluate the logarithm using technology. [Logarithms as solutions for exponentials]
4.1 Prove simple laws of logarithms. CA
4.2 Use the definition of logarithms to translate between logarithms in any base. CA
4.3 Understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their
approximate values. CA
I. Extend the domain of trigonometric functions using the unit circle.
1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
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 counterclockwise around the unit circle.
2.1 Graph all 6 basic trigonometric functions. CA
II. Model periodic phenomena with trigonometric functions.
5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
III. Prove and apply trigonometric identities.
8. Prove the Pythagorean identity sin2(θ ) + cos2(θ ) = 1 and use it to find sin(θ ), cos(θ ), or tan(θ ) given sin(θ ), cos(θ ),or tan(θ ) and the quadrant of the angle.