Precalculus serves as a bridge between Algebra 2 and Calculus, emphasizing functions, trigonometry, and advanced mathematical concepts. Here are the key learning goals:
1. Functions and Their Properties
- Understand the concept of a function, domain, range, and function notation.
- Analyze key features of functions (intercepts, intervals of increase/decrease, extrema, asymptotes).
- Perform operations on functions, including composition and inverses.
- Understand and apply transformations of functions (shifts, reflections, stretches, and compressions).
2. Polynomial and Rational Functions
- Perform operations with polynomials and use long and synthetic division.
- Solve polynomial equations using the Rational Root Theorem and Factoring Theorem.
- Analyze the behavior of polynomial functions (end behavior, turning points, and zeros).
- Understand and graph rational functions, identifying vertical, horizontal, and slant asymptotes.
3. Exponential and Logarithmic Functions
- Understand the properties of exponential growth and decay.
- Apply logarithm laws to simplify expressions and solve equations.
- Graph and analyze exponential and logarithmic functions.
- Solve real-world problems involving exponential and logarithmic functions (e.g., compound interest, population growth).
4. Trigonometry
- Understand unit circle definitions of sine, cosine, and tangent.
- Evaluate trigonometric functions at special angles and general angles.
- Graph sine, cosine, tangent, secant, cosecant, and cotangent functions, including transformations.
- Solve trigonometric equations using identities, inverse functions, and algebraic techniques.
- Apply trigonometric identities (Pythagorean, reciprocal, sum/difference, double/half-angle).
- Solve problems using the Laws of Sines and Cosines.
5. Vectors and Parametric Equations
- Understand vector operations (addition, subtraction, dot product, magnitude, direction).
- Represent motion using parametric equations and convert between parametric and Cartesian equations.
6. Polar Coordinates and Complex Numbers
- Convert between polar and rectangular coordinates.
- Graph polar equations and analyze their symmetries.
- Perform operations with complex numbers in polar form (multiplication, division, De Moivre’s Theorem).
7. Sequences and Series
- Analyze arithmetic and geometric sequences and series.
- Use formulas for nth terms and sums of series.
- Understand and apply the Binomial Theorem.
- Work with sigma notation and infinite series, including convergence and divergence.
8. Limits and Introductory Calculus Concepts
- Understand the concept of a limit and its notation.
- Estimate limits graphically and numerically.
- Recognize asymptotic and end behavior in terms of limits.
- Explore the concept of instantaneous rate of change as a precursor to derivatives.