Castle Learning

Pre-Calculus

Castle Learning Online's Pre-Calculus course features a wide-range of graphics and fully worked-out problem solutions incorporating both algebraic and calculator techniques. In addition to a comprehensive listing of formulas and equations, the extensive set of reference information provides step-by-step instructions and examples of many advanced algebraic techniques including completing the square, grouping, synthetic division, u-substitution, matrix arithmetic, and regressions. The Calculator Tips include step-by-step instructions and examples of many tasks commonly done on a graphing calculator, including finding the roots, maxima, minima, points of intersection and slopes of graphs; regressions; scatter plots and curves of best fit; matrix operations and solving matrix equations; and calculating the area under a curve. Reference tables included.

Units: (click to view Sections)

Functions
Sections:
1. Function Notation and Evaluation
2. Compositions and Decompositions
3. Algebra of Functions
4. Inverse Functions
5. Domain and Range (graph and equation)
6. Solving Equations (Radical, quadratic, absolute value, fractional, f(x) = 0)
Linear and Quadratic Functions
Sections:
1. The Equation of a Quadratic (parabola)
2. Parallel and Perpendicular Lines
3. Regressions (Calculator) and Word Problems
4. Completing the Square (Vertex form and finding roots)
5. Parabola and Linear Word Problems (max, min, zeros)
Polynomial Functions
Sections:
1. Synthetic Division
2. Solving Polynomial Equations using u-Substitution
3. Characteristics of Parabolas, Cubics, Quartics and Quintics
4. Relative and Absolute Maximum and/or Minimum and Range
5. Sketch given an Equation
6. Increasing and Decreasing Functions
7. Even and Odd Functions
8. Rational Functions
9. Radical Functions
Matrices
Sections:
1. Addition and Subtraction with Matrices
2. Multiplication with Matrices
3. Inverses
4. Solving Systems using Matrices (word problems)
5. Matrix Equations
Sequences and Series
Sections:
1. Sequences
2. Arithmetic and Geometric Sequences
3. Arithmetic and Geometric Series
4. Sums, Binomial Expansion, Binomial Theorem, Combinations, Pascal
5. Limits of Sequences
6. Sums of Infinite Series
7. Word Problems
Exponential and Logarithmic Functions
Sections:
1. Rational Exponents
2. Exponential and Logarithmic Functions and Regressions
3. Properties of Logarithms (expansion and contraction)
4. Exponential and Log Equations and Inequalities including Domain
5. Growth and Decay
6. Compound Interest
Graphing Functions with Transformations
Sections:
1. Vertex Form of a Parabola and Absolute Value
2. Logs and Exponentials
3. Radicals
4. Domain and Range
Trigonometric Functions
Sections:
1. Trigonometric Ratios of Acute Angles
2. Angle Measurement: Degrees and Radians
3. Extended Definitions of Trigonometric Ratios
4. Pythagorean and Quotient Identities
5. The Law of Cosines
6. The Law of Sines
7. The Area of a Triangle
8. The Sum and Difference of Two Angles
9. Double and Half Angle Identities
10. Trigonometric Equations
11. Graphing and Transformations
Parametric and Polar Equations
Sections:
1. Convert between Rectangular and Parametric Form
2. Convert between Rectangular and Polar Form
3. Graphing Equations
4. Vectors
5. Writing Complex Numbers in Polar Form
Limits and Derivatives
Sections:
1. Limit Definition of Derivatives
2. Derivatives (short cut) and Graphs of Derivatives
3. Applications (velocity, acceleration, max and min, Tangent Lines)
4. Area under a Curve
5. Limits
6. Points of Inflection

Themes:

Operations
Estimation
Variables and Expressions
Equations and Inequalities
Patterns, Relations and functions
Geometry (angles, lines)
Trigonometry
Area, Volume, Perimeter of shapes including polygons and circles
Transformational Geometry
Coordinate Geometry
Analysis of Data
Predictions from Data

Attributes:

Diagram
Table
Graph
Word Problem
Proof
Short Answer Fill-in