Pre-Calculus

Pre-Calculus prepares students for calculus by covering functions, trigonometry, and analytical geometry.

Functions and Their PropertiesTrigonometric FunctionsAnalytical Geometry
Polynomial and Rational FunctionsExponential and Logarithmic FunctionsConic Sections
Modeling Real-World DataEngineering and DesignFinance and Population Growth
Graphing TechniquesProblem-Solving and Test Preparation
Functions and Their ...Trigonometric Functi...Analytical GeometryPolynomial and Ratio...Exponential and Loga...Conic SectionsModeling Real-World ...Engineering and Desi...Finance and Populati...Graphing TechniquesProblem-Solving and ...
Advanced Topics

Polynomial and Rational Functions

Diving Deeper: Polynomial and Rational Functions

Polynomial functions involve terms like \( x^2, x^3, \) and so on. They can model a wide variety of curves, depending on their degree and coefficients. Rational functions are ratios of two polynomials, like \( f(x) = \frac{x^2 + 1}{x - 3} \).

Features

  • End Behavior: How the function acts as \( x \) gets very large or very small.
  • Zeros: Where the function crosses the \( x \)-axis.
  • Asymptotes: Lines the graph approaches but never touches (more common in rational functions).

Why They Matter

These functions model everything from projectile motion to complex economic systems.

Key Formula

\[f(x) = \frac{P(x)}{Q(x)}\]

Examples

  • Calculating the profit of a business over time with a cubic polynomial.

  • Modeling the speed of a car as it approaches a speed limit using a rational function.

In a Nutshell

Polynomial and rational functions extend the idea of functions to more complex algebraic expressions.

Key Terms

Asymptote
A line that a graph approaches but never actually touches.
Zero
An input value that makes the function output zero.
Next
Exponential and Logarithmic Functions