Module 1

Functions

Functions are the foundation of Calculus and much of the mathematics you have studied up to this point.  Having a solid foundation of functions is going to serve you well as you expand upon your education after Pre-Calculus.  Functions allow us to see the real world by interpreting and creating models that help us better understand and predict based on patterns.  Pre-Calculus can be taught out of order but functions is a necessary chapter that will serve you for the rest of the course.

Module Highlights:

• Characteristics of Functions - Understand what makes up a function and what the properties of a function tell us about the graph
• Implied Domain - Interpret the meaning of values that make a function undefined and defined by writing out the domain in interval notation
• Transformations of Functions - Move functions around based on the rules assigned to them by their equation to graph and identify the domain
• Piecewise Functions - Learn how to graph more that one function on a coordinate plane using transformations and domain restrictions
• Operations - Explain what happens when we add, subtract, multiply and divide functions and determine the domain
• Inverse of Functions - Make clear the relationship between a function and its inverse algebraically and graphically

Module 4

Trigonometry

Trigonometry is like no other topic we have covered up to this point but we can apply what we previously learned in our unit studying triangles and the relationships of their sides and angles.  In this chapter we will focus on right triangles and how to understand and apply the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent.

Module Highlights:

Curiosity-driven fascinations breaking down the process of how they’ll achieve the promise of the module.

• Right Triangles - Find the missing sides of Right Triangles using the Pythagorean Theorem as well as trigonometric identities SOHCAHTOA
• Angles in Trigonometry - Develop the understanding of an angle in standard form and how the idea of radians, degrees minutes and seconds can be applied
• Angles and Constraints - Understand how constraints can affect the trigonometric ratios when being applied on a coordinate grid
• The Unit Circle - Use your understanding of right triangles, trigonometric ratios and constraints to develop and apply the unit circle
• Graph Sine, Cosine and Tangent - Identify the key components of what makes a trigonometric graph and how those relate to the values on a coordinate grid and unit circle
• Graph Cosecant, Secant, and Cotangent - Make clear the relationship between the graphs of trigonometric functions and their reciprocal counterpart
• Trigonometric Inverses - Apply constraints of functions to understand when to evaluate the inverse of a trigonometric function as well as the properties of its graph

Module 5

Analytic Trigonometry

Analytic Trigonometry takes everything that we have learned in our last chapter and begins to apply our understanding to dive deeper into how the six trigonometric functions are related to one another and with algebra.  Throughout this chapter students will be required to refresh their algebra skills and apply them with trigonometric terms.

Module Highlights: 

• Simplify Identities - Simplify trigonometric expressions to a single number or function by using basic trigonometric identities and algebraic techniques
• Verify Identities - Apply trigonometric identities as well as algebraic techniques to simplify one side of an identity to verify it is the same as the other
• Solve Trigonometric Equations - Find the solutions to a trigonometric equation using algebraic techniques, trigonometric identities and our understanding of the unit circle as well as the graphs of each trigonometric function
• Sum and Difference Formulas - Learn how to simplify, verify, solve and evaluate using the sum and difference formulas for sine, cosine and tangent

• Double and Half Angle Formulas - Learn how to simplify, verify, solve and evaluate using the sum and difference formulas for sine, cosine and tangent

Module 6

Applications of Trigonometry

Applications of trigonometry moves our understanding of trigonometry into the real word with applications that will help students explore the possibilities of what they have been learning through multiple word problems in each chapter.

Module Highlights:

• Word Problems in Trigonometry - Apply basic evaluating trigonometric functions and their inverse functions to find the missing side of a right triangle given a word problem
• Law of Sines - Explore how to find the missing sides of a triangle when given an oblique triangle and when the information provided gives the ambiguous case
• Law of Cosines - Use the law of sines and cosines to find all the missing sides of an oblique triangle as well as determine the area of an oblique triangle
• Vectors on a Plane - Learn the basics of vector operations such as graphing, magnitude, and the unit vector to better understand the importance of vectors
• Applications of Vectors - Apply your understanding of vectors and their different forms to solve word problems using vectors as well as apply the dot product
• Parametric Equations - Take previously understood functions and describe them as separate functions based on a variable called the parameter
• Polar Equations - Develop a new understanding for points and equations on a coordinate grid using the angle and distance from the origin

Module 7

Conic Sections

Conic sections expand on the lessons from parametric and polar equations to understand equations that are not functions.  We do this by understanding the graphs of conic sections and different ways we can slice a cone.

Module Highlights: 

• Classifying Conic Sections - Classify the type of conic sections given the equation in standard and general form by completing the square and inspection
• The Parabola - Determine the parts of a parabola to help sketch its graph as well as write the equation in standard and general form
• The Ellipse - Determine the parts of an ellipse to help sketch its graph as well as write the equation in standard and general form
• The Hyperbola - Determine the parts of a hyperbola to help sketch its graph as well as write the equation in standard and general form

Module 8

The Limit

The limit is our introduction into Calculus.  We will spend a little bit of time understanding the idea and purpose of the limit but mainly focus our attention on the different methods to evaluate the limit based on the techniques we have learned in this course.

Module Highlights:

• The General Limit - Develop your understanding of the general limit by using a table of values, direct substitution and a graph
• Left and Right Hand Limits - Use your understanding of the general limit to further evaluate the left and right hand limits algebraically and from a graph
• Limits at Infinity - Apply your understanding of vertical and horizontal asymptotes and end behavior to evaluate the limit at positive and negative infinity