Graphing Rational Functions and Transformations: Study Notes for College Algebra

Graphing Rational Functions Using Transformations

Introduction to Rational Function Transformations

Rational functions can be graphed by applying transformations to the basic function $f(x) = \frac{1}{x}$. 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 $h$ and $k$ in $g(x) = -f(-x-h) + k$.

• General Transformation Formula:

$g(x) = a \cdot f(b(x-h)) + k$

• 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:

1. Plot Asymptotes: Identify and plot vertical and horizontal asymptotes.

2. Reflect: Apply reflections over axes as indicated by the transformation.

3. Shift: Move the graph horizontally and vertically according to $h$ and $k$.

4. Sketch Curves: Draw the graph approaching the asymptotes.

Example 1

Graph $g(x) = \frac{1}{x-1} + 3$

• Vertical Asymptote: $x = 1$

• Horizontal Asymptote: $y = 3$

• Shift: Right by 1, up by 3

• Domain: $(-\infty, 1) \cup (1, \infty)$

• Range: $(-\infty, 3) \cup (3, \infty)$

Example 2

Graph $g(x) = \frac{1}{(x+3)^2} - 2$

• Vertical Asymptote: $x = -3$

• Horizontal Asymptote: $y = -2$

• Shift: Left by 3, down by 2

• Domain: $(-\infty, -3) \cup (-3, \infty)$

• Range: $(-2, \infty)$

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 $x$.

• 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 $x$.

• y-intercept: Evaluate $f(0)$.

Step-by-Step Graphing Procedure

1. Factor and Find Domain: Set denominator equal to zero to find excluded values.

2. Identify Holes: Set common factors in numerator and denominator equal to zero.

3. Find x-intercepts and Behavior: Set numerator equal to zero and analyze multiplicity.

4. Find y-intercept: Compute $f(0)$.

5. Find Vertical Asymptotes: Set denominator equal to zero.

6. Find Horizontal/Slant Asymptotes: Compare degrees of numerator and denominator.

7. Plot Points: Choose intervals between asymptotes and intercepts, plot sample points.

8. Draw Curves: Connect points, approaching asymptotes.

Example: Graph $f(x) = \frac{2x}{x-3}$

• Vertical Asymptote: $x = 3$

• Horizontal Asymptote: $y = 2$ (since degrees are equal, divide leading coefficients)

• x-intercept: $x = 0$

• y-intercept: $f(0) = 0$

• Domain: $x \neq 3$

Table: Summary of Rational Function Features

Feature	How to Find	Example
Vertical Asymptote	Set denominator = 0	$x = 3$ for $f(x) = \frac{2x}{x-3}$
Horizontal Asymptote	Compare degrees; divide leading coefficients if equal	$y = 2$ for $f(x) = \frac{2x}{x-3}$
Hole	Common factor cancels in numerator and denominator	None in $f(x) = \frac{2x}{x-3}$
x-intercept	Set numerator = 0	$x = 0$
y-intercept	Evaluate $f(0)$	$f(0) = 0$

Practice Problems

Practice: Graphing Rational Functions Using Transformations

• Problem A: $g(x) = \frac{1}{x-1} + 3$

• Problem B: $g(x) = \frac{1}{(x+3)^2} - 2$

• Follow the step-by-step procedure above to graph each function, identifying asymptotes, intercepts, and shifts.

Practice: Graphing General Rational Functions

• Problem: $f(x) = \frac{x^2 + 3}{x^2 + 5x + 6}$

• 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.