### MM4A1: Students will explore rational functions.

#### MM4A1.a: Investigate and explain characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior.

General Form of a Rational Function

Rational Functions

#### MM4A1.b: Find inverses of rational functions, discussing domain and range, symmetry, and function composition.

Function Machines 3 (Functions and Problem Solving)

General Form of a Rational Function

Rational Functions

#### MM4A1.c: Solve rational equations and inequalities analytically, graphically, and by using appropriate technology.

Solving Linear Inequalities using Addition and Subtraction

Solving Linear Inequalities using Multiplication and Division

### MM4A2: Students will use the circle to define the trigonometric functions.

#### MM4A2.a: Define and understand angles measured in degrees and radians, including but not limited to 0°, 30°, 45°, 60°, 90°, their multiples, and equivalences.

Cosine Function

Sine Function

Tangent Function

#### MM4A2.b: Understand and apply the six trigonometric functions as functions of general angles in standard position.

Cosine Function

Sine Function

Tangent Function

#### MM4A2.c: Find values of trigonometric functions using points on the terminal sides of angles in the standard position.

Cosine Function

Sine Function

Tangent Function

#### MM4A2.d: Understand and apply the six trigonometric functions as functions of arc length on the unit circle.

Cosine Function

Sine Function

Tangent Function

Unit Circle

#### MM4A2.e: Find values of trigonometric functions using the unit circle.

Cosine Function

Sine Function

Tangent Function

Unit Circle

### MM4A3: Students will investigate and use the graphs of the six trigonometric functions.

#### MM4A3.a: Understand and apply the six basic trigonometric functions as functions of real numbers.

Cosine Function

Sine Function

Tangent Function

Unit Circle

#### MM4A3.b: Determine the characteristics of the graphs of the six basic trigonometric functions.

Cosine Function

Sine Function

Tangent Function

Unit Circle

#### MM4A3.c: Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift.

Cosine Function

Sine Function

Tangent Function

Translating and Scaling Functions

Translating and Scaling Sine and Cosine Functions - Activity A

#### MM4A3.d: Apply graphs of trigonometric functions in realistic contexts involving periodic phenomena.

Cosine Function

Sine Function

Tangent Function

Translating and Scaling Sine and Cosine Functions - Activity A

Unit Circle

### MM4A4: Students will investigate functions.

#### MM4A4.a: Compare and contrast properties of functions within and across the following types: linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric, and piecewise.

Exponential Functions - Activity A

Function Machines 2 (Functions, Tables, and Graphs)

General Form of a Rational Function

Logarithmic Functions - Activity A

Logarithmic Functions: Translating and Scaling

Quadratics in Factored Form

Rational Functions

Roots of a Quadratic

Sine Function

Tangent Function

#### MM4A4.b: Investigate transformations of functions.

Translating and Scaling Functions

#### MM4A4.c: Investigate characteristics of functions built through sum, difference, product, quotient, and composition.

Addition and Subtraction of Polynomials

### MM4A5: Students will establish the identities below and use them to simplify trigonometric expressions and verify equivalence statements.

#### MM4A5.1: tan theta = sin theta/cos theta

Sine, Cosine and Tangent

Tangent Function

Tangent Ratio

#### MM4A5.3: sec theta = 1/cos theta

Simplifying Trigonometric Expressions

#### MM4A5.4: csc theta = 1/sin theta

Simplifying Trigonometric Expressions

#### MM4A5.5: sin² theta + cos² theta = 1

Simplifying Trigonometric Expressions

#### MM4A5.6: 1 + tan² theta = sec² theta

Simplifying Trigonometric Expressions

#### MM4A5.7: 1 + cot² theta = csc² theta

Simplifying Trigonometric Expressions

#### MM4A5.8: sin (alpha ± beta)= sin alpha cos beta ± cos alpha sin beta

Sum and Difference Identities for Sine and Cosine

#### MM4A5.9: cos (alpha ± beta) = cos alpha cos beta ± sin alpha sin beta

Sum and Difference Identities for Sine and Cosine

#### MM4A5.10: sin (2 theta) = 2 sin theta cos theta

Sum and Difference Identities for Sine and Cosine

#### MM4A5.11: cos (2 theta) = cos² theta - sin² theta

Sum and Difference Identities for Sine and Cosine

### MM4A6: Students will solve trigonometric equations both graphically and algebraically.

#### MM4A6.b: Use the coordinates of a point on the terminal side of an angle to express x as r cos theta and y as r sin theta.

Points in Polar Coordinates

### MM4A9: Students will use sequences and series

#### MM4A9.a: Use and find recursive and explicit formulae for the terms of sequences.

Arithmetic Sequences

Arithmetic and Geometric Sequences

Geometric Sequences

#### MM4A9.b: Recognize and use simple arithmetic and geometric sequences.

Arithmetic Sequences

Arithmetic and Geometric Sequences

Geometric Sequences

### MM4A10: Students will understand and use vectors.

#### MM4A10.a: Represent vectors algebraically and geometrically.

Adding Vectors

Vectors

#### MM4A10.b: Convert between vectors expressed using rectangular coordinates and expressed using magnitude and direction.

Adding Vectors

Vectors

#### MM4A10.c: Add, subtract, and compute scalar multiples of vectors.

Adding Vectors

Vectors

#### MM4A10.d: Use vectors to solve realistic problems.

Adding Vectors

Vectors

### MM4D1: Using simulation, students will develop the idea of the central limit theorem.

Probability Simulations

### MM4D2: Using student-generated data from random samples of at least 30 members, students will determine the margin of error and confidence interval for a specified level of confidence.

Polling: City

Polling: Neighborhood

### MM4D3: Students will use confidence intervals and margins of error to make inferences from data about a population. Technology is used to evaluate confidence intervals, but students will be aware of the ideas involved.

Polling: City

Correlation last revised: 9/11/2014