Graphing Rational Functions: Transformations and Asymptotes

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 $f(x) = \frac{1}{x}$. Transformations include reflections, shifts, and stretches/compressions.

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

• Horizontal Shift: $h$ units right ($y = f(x-h)$) or left ($y = f(x+h)$).

• Vertical Shift: $k$ units up ($y = f(x) + k$) or down ($y = f(x) - k$).

• General Transformation Formula:

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

• Example: For $f(x) = \frac{1}{x}$, $g(x) = \frac{1}{x-2} + 3$ 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 $f(x) = \frac{1}{x}$.

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

• Vertical Asymptote: $x = 2$

• Horizontal Asymptote: $y = 3$

• Domain: $x \neq 2$

• Range: $y \neq 3$

Practice: Graphing Rational Functions with Transformations

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

• Example (A): $g(x) = \frac{1}{x+3} - 2$

• Vertical Asymptote: $x = -3$

• Horizontal Asymptote: $y = -2$

• Domain: $x \neq -3$

• Range: $y \neq -2$

• Example (B): $g(x) = \frac{2}{x^2}$

• Vertical Asymptote: $x = 0$

• Horizontal Asymptote: $y = 0$

• Domain: $x \neq 0$

• Range: $y > 0$

How to Graph Rational Functions

Step-by-Step Procedure

Graphing a general rational function $f(x) = \frac{P(x)}{Q(x)}$ involves several key steps:

1. Factor Numerator and Denominator: Write $P(x)$ and $Q(x)$ in factored form.

2. Find Vertical Asymptotes: Set $Q(x) = 0$ and solve for $x$ (values where the function is undefined).

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

4. Find x-intercepts: Set $P(x) = 0$ and solve for $x$ (where the graph crosses the x-axis).

5. Find y-intercept: Evaluate $f(0)$.

6. Find Horizontal/Oblique Asymptotes:

   • If degree of $P(x)$ < degree of $Q(x)$, horizontal asymptote at $y = 0$.

   • If degrees are equal, horizontal asymptote at $y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}$.

   • If degree of $P(x)$ > degree of $Q(x)$, 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: $f(x) = \frac{x^2 - 2x}{x^2 - 1}$

• Factor: Numerator: $x(x-2)$, Denominator: $(x-1)(x+1)$

• Vertical Asymptotes: $x = 1$, $x = -1$

• x-intercepts: $x = 0$, $x = 2$

• y-intercept: $f(0) = 0$

• Horizontal Asymptote: Degrees equal, so $y = 1$

• Domain: $x \neq 1, x \neq -1$

• Range: $y \neq 1$

Practice: Graphing Another Rational Function

• Given: $f(x) = \frac{x+1}{x^2 - 5x + 6}$

• Factor: Denominator: $(x-2)(x-3)$

• Vertical Asymptotes: $x = 2$, $x = 3$

• x-intercept: $x = -1$

• y-intercept: $f(0) = \frac{1}{6}$

• Horizontal Asymptote: Degree numerator < denominator, so $y = 0$

• Domain: $x \neq 2, x \neq 3$

Summary Table: Key Features of Rational Functions

Feature	How to Find	Example
Vertical Asymptote	Set denominator = 0	$x = 2$ for $f(x) = \frac{1}{x-2}$
Horizontal Asymptote	Compare degrees	$y = 0$ for $f(x) = \frac{1}{x^2}$
x-intercept	Set numerator = 0	$x = 0$ for $f(x) = \frac{x}{x-1}$
y-intercept	Evaluate $f(0)$	$y = 1$ for $f(x) = \frac{1}{x-2} + 1$
Hole	Common factor cancels	$x = 1$ for $f(x) = \frac{(x-1)}{(x-1)(x+2)}$

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.