Graphing Rational Functions

Transformations of Rational Functions

Rational functions can be graphed by applying transformations to basic parent functions. The most common parent rational function is . Transformations include reflections, shifts, and stretches/compressions.

• Reflection: Over the x-axis () or y-axis ().

• Horizontal Shift: units right () or left ().

• Vertical Shift: units up () or down ().

• General Transformation Formula:

• Example: For , shifts the graph 2 units right and 3 units up.

Graphing Rational Functions Using Transformations

To graph a transformed rational function, follow these steps:

1. Plot the parent function .

2. Apply the indicated transformations (shifts, reflections, stretches).

3. Draw the new asymptotes according to the shifts.

4. Sketch the graph, noting the behavior near asymptotes.

• Example:

• Vertical Asymptote:

• Horizontal Asymptote:

• Domain:

• Range:

Practice: Graphing Rational Functions with Transformations

Practice problems involve identifying the transformations and sketching the resulting graph.

• Example (A):

• Vertical Asymptote:

• Horizontal Asymptote:

• Domain:

• Range:

• Example (B):

• Vertical Asymptote:

• Horizontal Asymptote:

• Domain:

• Range:

How to Graph Rational Functions

Step-by-Step Procedure

Graphing a general rational function involves several key steps:

1. Factor Numerator and Denominator: Write and in factored form.

2. Find Vertical Asymptotes: Set and solve for (values where the function is undefined).

3. Find Holes: If a factor cancels in both numerator and denominator, there is a hole at that -value.

4. Find x-intercepts: Set and solve for (where the graph crosses the x-axis).

5. Find y-intercept: Evaluate .

6. Find Horizontal/Oblique Asymptotes:

   • If degree of < degree of , horizontal asymptote at .

   • If degrees are equal, horizontal asymptote at .

   • If degree of > degree of , use polynomial division for oblique asymptote.

7. Plot Key Points and Sketch: Draw the asymptotes, plot intercepts, and sketch the graph, showing behavior near asymptotes.

Example: Graphing a Rational Function

• Given:

• Factor: Numerator: , Denominator:

• Vertical Asymptotes: ,

• x-intercepts: ,

• y-intercept:

• Horizontal Asymptote: Degrees equal, so

• Domain:

• Range:

Practice: Graphing Another Rational Function

• Given:

• Factor: Denominator:

• Vertical Asymptotes: ,

• x-intercept:

• y-intercept:

• Horizontal Asymptote: Degree numerator < denominator, so

• Domain:

Summary Table: Key Features of Rational Functions

Additional info: These notes expand on the original slides by providing definitions, step-by-step procedures, and a summary table for rational function graphing, suitable for Precalculus students.