1: Arithmetic with Polynomials and Rational Expressions

1.1: Know and apply the Division Theorem and the Remainder Theorem for polynomials.

Dividing Polynomials Using Synthetic Division

1.2: Graph polynomials identifying zeros when suitable factorizations are available and indicating end behavior. Write a polynomial function of least degree corresponding to a given graph.

Modeling the Factorization of x²+bx+c
Polynomials and Linear Factors

1.3: Prove polynomial identities and use them to describe numerical relationships.

Factoring Special Products

2: Reasoning with Equations and Inequalities

2.1: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions.

Solving Equations by Graphing Each Side
Solving Linear Systems (Matrices and Special Solutions)
Solving Linear Systems (Slope-Intercept Form)
Solving Linear Systems (Standard Form)

2.2: Represent a system of linear equations as a single matrix equation in a vector variable.

Solving Linear Systems (Matrices and Special Solutions)

2.3: Using technology for matrices of dimension 3 × 3 or greater, find the inverse of a matrix if it exists and use it to solve systems of linear equations.

Solving Linear Systems (Matrices and Special Solutions)

2.4: Solve an equation of the form f(x)=g(x) graphically by identifying the 𝑥𝑥-coordinate(s) of the point(s) of intersection of the graphs of y = f(x and y = g(x).

Point-Slope Form of a Line
Solving Equations by Graphing Each Side
Solving Linear Systems (Matrices and Special Solutions)
Solving Linear Systems (Slope-Intercept Form)
Standard Form of a Line

3: Structure and Expressions

3.1: Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions.

Compound Interest
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II

3.2: Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions.

Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II

4: Building Functions

4.1: Write a function that describes a relationship between two quantities.

4.1.1: Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations.

Addition and Subtraction of Functions
Function Machines 1 (Functions and Tables)
Function Machines 3 (Functions and Problem Solving)

4.2: Describe the effect of the transformations kf (x), f(x) +k, f(x + k), and combinations of such transformations on the graph of y = f(x) for any real number k. Find the value of k given the graphs and write the equation of a transformed parent function given its graph.

Absolute Value with Linear Functions
Exponential Functions
Introduction to Exponential Functions
Rational Functions
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions
Translations
Zap It! Game

4.3: Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, as f and g are inverse functions if and only if f(x)=y and g(y)=x, for all values of x in the domain of f and all values of y in the domain of g, and find inverse functions for one-to-one function or by restricting the domain.

4.3.2: If a function has an inverse, find values of the inverse function from a graph or table.

Function Machines 3 (Functions and Problem Solving)
Logarithmic Functions

4.4: Understand and verify through function composition that exponential and logarithmic functions are inverses of each other and use this relationship to solve problems involving logarithms and exponents.

Logarithmic Functions

5: Interpreting Functions

5.1: Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity.

Absolute Value with Linear Functions
Compound Interest
Exponential Functions
Function Machines 1 (Functions and Tables)
Function Machines 3 (Functions and Problem Solving)
General Form of a Rational Function
Graphs of Polynomial Functions
Introduction to Exponential Functions
Introduction to Functions
Logarithmic Functions
Points, Lines, and Equations
Quadratics in Factored Form
Quadratics in Polynomial Form
Radical Functions

5.2: Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes.

Introduction to Functions
Logarithmic Functions
Radical Functions

5.3: Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context.

Cat and Mouse (Modeling with Linear Systems)
Slope

5.4: Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases.

5.4.4: Graph trigonometric functions, showing period, midline, and amplitude.

Cosine Function
Sine Function
Tangent Function
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions

7: Trigonometry

7.2: Define sine and cosine as functions of the radian measure of an angle in terms of the x - and y - coordinates of the point on the unit circle corresponding to that angle and explain how these definitions are extensions of the right triangle definitions.

7.2.1: Define the tangent, cotangent, secant, and cosecant functions as ratios involving sine and cosine.

Simplifying Trigonometric Expressions
Sine, Cosine, and Tangent Ratios

7.2.2: Write cotangent, secant, and cosecant functions as the reciprocals of tangent, cosine, and sine, respectively.

Simplifying Trigonometric Expressions
Tangent Function

7.3: 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.

Cosine Function
Sine Function
Sum and Difference Identities for Sine and Cosine
Tangent Function
Translating and Scaling Sine and Cosine Functions

7.4: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Cosine Function
Sine Function
Tangent Function

7.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions

7.8: Justify the Pythagorean, even/odd, and cofunction identities for sine and cosine using their unit circle definitions and symmetries of the unit circle and use the Pythagorean identity to find sinA, cosA, or tanA, given sinA, cosA, or tanA, and the quadrant of the angle.

Cosine Function
Sine Function

7.9: Justify the sum and difference formulas for sine, cosine, and tangent and use them to solve problems.

Sum and Difference Identities for Sine and Cosine

9: Expressing Geometric Properties with Equations

9.1: Use the geometric definition of a parabola to derive its equation given the focus and directrix.

Parabolas

9.2: Use the geometric definition of an ellipse and of a hyperbola to derive the equation of each given the foci and points whose sum or difference of distance from the foci are constant.

Ellipses
Hyperbolas

11: Complex Number System

11.2: Find the conjugate of a complex number in rectangular and polar forms and use conjugates to find moduli and quotients of complex numbers.

Points in the Complex Plane
Roots of a Quadratic

11.3: Graph complex numbers on the complex plane in rectangular and polar form and explain why the rectangular and polar forms of a given complex number represent the same number.

Points in the Complex Plane

11.4: Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

Points in the Complex Plane

12: Vector and Matrix Quantities

12.1: 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.

Adding Vectors
Vectors

12.2: Represent and model with vector quantities. Use the coordinates of an initial point and of a terminal point to find the components of a vector.

Adding Vectors
Vectors

12.4: Perform operations on vectors.

12.4.1: Add and subtract vectors using components of the vectors and graphically.

Adding Vectors
Vectors

12.4.2: Given the magnitude and direction of two vectors, determine the magnitude of their sum and of their difference.

Vectors

12.7: Perform operations with matrices of appropriate dimensions including addition, subtraction, and scalar multiplication.

Translations

12.11: Apply 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Dilations
Translations