BackRational Functions, Their Graphs, and Polynomial & Rational Inequalities
Study Guide - Smart Notes
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3.5 Rational Functions & Their Graphs
Key Terms and Concepts
Rational Function: A function of the form , where and are polynomials and .
Domain: The set of all real numbers for which .
Finding the Domain of Rational Functions
Set the denominator equal to zero and solve for to find values excluded from the domain.
Example: For , set is excluded. Domain:
General Rational Functions
Graphs of rational functions often have vertical and horizontal asymptotes.
Vertical Asymptote: Occurs at where and .
Horizontal Asymptote: Determined by the degrees of and .
Asymptote Relations
Case | Horizontal Asymptote |
|---|---|
deg < deg | |
deg = deg | (ratio of leading coefficients) |
deg > deg | No horizontal asymptote (may have slant asymptote) |
Transforming Rational Functions
Basic rational function:
Transformations include shifts, reflections, and stretches/compressions.
Example: is shifted 2 units right.
Graphing Rational Functions
Identify vertical and horizontal asymptotes.
Plot intercepts and additional points as needed.
Sketch the graph, showing behavior near asymptotes.
Identifying and Analyzing Asymptotes
Vertical Asymptote: Set denominator to zero and solve for .
Horizontal Asymptote: Compare degrees of numerator and denominator.
Example: Vertical asymptote at ; horizontal asymptote at .
3.6 Polynomial & Rational Inequalities
Solving Polynomial Inequalities
Set the inequality to zero: or .
Factor the polynomial and find zeros.
Test intervals between zeros to determine where the inequality holds.
Example: Solve Factor: Solution: or
Solving Rational Inequalities
Write the inequality in the form .
Find zeros of numerator and denominator.
Mark these points on a number line and test intervals.
Exclude points where denominator is zero (undefined).
Example: Solve Zeros: (numerator), (denominator) Test intervals: , , Solution: (undefined at ),
Summary Table: Asymptote Identification
Asymptote Type | How to Find |
|---|---|
Vertical | Set denominator |
Horizontal | Compare degrees of numerator and denominator |
Slant (Oblique) | Degree numerator degree denominator ; use polynomial division |
Key Points
Always check for extraneous solutions, especially when multiplying or dividing by expressions involving variables.
Graphical analysis can help verify algebraic solutions.
Example Problems
Find the domain: Solution:
Solve: Solution: or
Solve: Solution: (undefined at ),
Additional info: These notes cover the core concepts of rational functions, their graphs, and solving polynomial and rational inequalities, which are essential topics in a Precalculus course. The examples and tables provided are inferred and expanded for clarity and completeness.
