Graphing Rational Functions and Transformations: Study Notes for College Algebra

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Graphing Rational Functions Using Transformations

Introduction to Rational Function Transformations

Rational functions can be graphed by applying transformations to the basic function . Transformations include reflections, shifts, and stretches, which alter the position and orientation of the graph.

Reflection: Over the x-axis or y-axis changes the sign of the function or its input.
Horizontal and Vertical Shifts: Moving the graph left/right or up/down by adjusting and in .
General Transformation Formula:

Vertical Asymptote: Occurs where the denominator is zero.
Horizontal Asymptote: Determined by the degrees of numerator and denominator.

Step-by-Step Graphing of Transformed Rational Functions

To graph a transformed rational function, follow these steps:

Plot Asymptotes: Identify and plot vertical and horizontal asymptotes.
Reflect: Apply reflections over axes as indicated by the transformation.
Shift: Move the graph horizontally and vertically according to and .
Sketch Curves: Draw the graph approaching the asymptotes.

Example 1

Graph

Vertical Asymptote:
Horizontal Asymptote:
Shift: Right by 1, up by 3
Domain:
Range:

Example 2

Graph

Vertical Asymptote:
Horizontal Asymptote:
Shift: Left by 3, down by 2
Domain:
Range:

How to Graph Rational Functions

Key Features of Rational Functions

When graphing rational functions, it is essential to identify vertical and horizontal asymptotes, x- and y-intercepts, and holes. These features determine the overall shape and position of the graph.

Vertical Asymptotes: Set denominator equal to zero and solve for .
Horizontal Asymptotes: Compare degrees of numerator and denominator.
Holes: Occur where a factor cancels in both numerator and denominator.
x-intercepts: Set numerator equal to zero and solve for .
y-intercept: Evaluate .

Step-by-Step Graphing Procedure

Factor and Find Domain: Set denominator equal to zero to find excluded values.
Identify Holes: Set common factors in numerator and denominator equal to zero.
Find x-intercepts and Behavior: Set numerator equal to zero and analyze multiplicity.
Find y-intercept: Compute .
Find Vertical Asymptotes: Set denominator equal to zero.
Find Horizontal/Slant Asymptotes: Compare degrees of numerator and denominator.
Plot Points: Choose intervals between asymptotes and intercepts, plot sample points.
Draw Curves: Connect points, approaching asymptotes.

Example: Graph

Vertical Asymptote:
Horizontal Asymptote: (since degrees are equal, divide leading coefficients)
x-intercept:
y-intercept:
Domain:

Table: Summary of Rational Function Features

Feature	How to Find	Example
Vertical Asymptote	Set denominator = 0	for
Horizontal Asymptote	Compare degrees; divide leading coefficients if equal	for
Hole	Common factor cancels in numerator and denominator	None in
x-intercept	Set numerator = 0
y-intercept	Evaluate

Practice Problems

Practice: Graphing Rational Functions Using Transformations

Problem A:
Problem B:
Follow the step-by-step procedure above to graph each function, identifying asymptotes, intercepts, and shifts.

Practice: Graphing General Rational Functions

Problem:
Factor denominator, find domain, identify holes, intercepts, and asymptotes, then sketch the graph.

Additional info: These notes expand on the original slides by providing full definitions, step-by-step procedures, and a summary table for rational function features. All equations are formatted in LaTeX for clarity.