Page 2: Understanding Horizontal Asymptotes

This page details the process of finding horizontal asymptotes through degree comparison between numerator and denominator polynomials.

Definition: Horizontal asymptotes are determined by comparing the degrees of numerator $n$ and denominator $d$ polynomials.

Highlight: There can only be one horizontal asymptote maximum for any rational function.

Example: For f$x$ = x-2$$x-1/x-2$$x-4, after cancellation, the horizontal asymptote is y = 1.

Quote: "If you big on top, just stop!" - referring to when the numerator degree exceeds the denominator degree.

Page 3: Systematic Approach to Graphing Rational Functions

This page outlines a comprehensive seven-step process for graphing rational functions effectively.

Highlight: The systematic approach includes factoring, finding holes, intercepts, asymptotes, and sketching branches.

Example: For f$x$ = x²+3x+2/x²+x-2, the process reveals a hole at (2,3), x-intercept at $-1,0$, and vertical asymptote at x = 1.

Definition: A hole occurs in a rational function when the same factor cancels in both numerator and denominator.

Page 4: Advanced Examples and Applications

This page provides multiple complex examples demonstrating the complete process of analyzing and graphing rational functions.

Example: The function f$x$ = $5x-10$/$x²+x-6$ is analyzed showing a hole at (2,1), vertical asymptote at x = -3, and horizontal asymptote at y = 0.

Highlight: Each example demonstrates different combinations of features including holes, asymptotes, and intercepts.

Vocabulary: DNE (Does Not Exist) is used to indicate when certain features are not present in a rational function.

Page 1: Introduction to Rational Functions and Asymptotes

This page introduces the fundamental concepts of rational functions and their key components. A rational function is explained as a ratio of two polynomials, with emphasis on branches and asymptotes.

Definition: A rational function is expressed as f$x$/g$x$ where f$x$ and g$x$ are polynomials.

Highlight: A rational function has one more branch than the number of vertical asymptotes.

Example: The function y = $x-3$/$x²-9$ demonstrates vertical asymptotes at x = -3 and x = 3.

Vocabulary: Branches refer to the continuous portions of a rational function graph separated by vertical asymptotes.