Pre-Calculus Overview and Functions

Study Notes

Pre-calculus is a course that prepares students for calculus.
It builds upon algebra, geometry, and trigonometry.
It covers topics like functions, graphs, limits, and other foundational concepts.

A function is a relation where each input has exactly one output.
Functions can be represented by equations, tables, graphs, or verbal descriptions.
Important types of functions include linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric functions.
Key characteristics of functions include domain, range, intercepts, maximum/minimum values, and asymptotes.
Understanding function transformations (shifts, stretches, reflections) is essential.

Graphs visually represent functions and relations.
Key features of graphs include intercepts, slopes, concavity, and asymptotes.
Graphing techniques include plotting points, using transformations, and recognizing standard graph shapes.
Different types of graphs like scatter plots and other types of statistical charts exist

Limits describe the behavior of a function as the input approaches a certain value.
Understanding limits is crucial for calculus, especially for evaluating derivatives and integrals.
Methods used include direct substitution, factoring, rationalizing the numerator, and using L'Hôpital's rule in appropriate situations.

Trigonometry deals with the relationships between angles and sides of triangles.
Key trigonometric functions include sine, cosine, tangent, cosecant, secant, and cotangent.
Concepts like trigonometric identities, inverse trigonometric functions, and trigonometric equations.
Graphs of trigonometric functions are also important, and their characteristics.

Polynomial functions are functions that are defined by expressions with sums of terms with variables raised to non-negative integer powers.
Understanding the behavior of polynomials, like end behavior and zeros (or roots) of the polynomial.
Fundamental Theorem of Algebra applies to finding the roots of a polynomial function.
Polynomial Graphs are fundamental to understanding the function.

Exponential functions involve exponents, and understanding their properties is crucial.
Exponential growth and decay are important applications.
Logarithmic functions are the inverse of exponential functions.
Properties of logarithms are essential for simplifying and solving equations.

Solving systems of equations involves finding the values that satisfy multiple equations simultaneously.
Linear systems can be solved by graphing, substitution, or elimination.
Non-linear systems involve more complex methods for solutions.
Different types of system equations exist, and students need to know when one method of solving a system will be most appropriate.