E6

Rational Functions and Asymptotes

Introduction to Rational Functions

Rational functions are quotients of two polynomials and are fundamental in modeling various real-world phenomena. Their behavior is often characterized by the presence of asymptotes, which describe the function's tendencies as the input grows large or approaches certain critical values.

• Definition: A rational function is any function of the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• Vertical Asymptotes: Occur at values of $x$ where $Q(x) = 0$ and $P(x) \neq 0$.

• Horizontal Asymptotes: Describe the end behavior as $x \to \infty$ or $x \to -\infty$.

• Holes: Occur at values of $x$ where both $P(x)$ and $Q(x)$ are zero, after simplification.

Identifying Asymptotes and Holes

To analyze rational functions, it is important to determine the location of vertical and horizontal asymptotes, as well as any holes in the graph.

• Vertical Asymptote Example: $f(x) = \frac{1}{(x-1)(x-3)}$ has vertical asymptotes at $x=1$ and $x=3$.

• Vertical Asymptote and Hole Example: $f(x) = \frac{1}{(x-1)(x-2)}$ has vertical asymptotes at $x=1$ and $x=2$; if the numerator also has $(x-2)$, a hole occurs at $x=2$.

Table: Types of Asymptotes in Rational Functions

Function	Vertical Asymptote	Horizontal Asymptote	Slant Asymptote	Other
$f(x) = \frac{1}{x-2}$	$x=2$	$y=0$	None	No Asymptotes
$f(x) = \frac{2x}{x-2}$	$x=2$	$y=2$	None	No Asymptotes
$f(x) = \frac{x^2}{x-2}$	$x=2$	None	Yes	No Asymptotes
$f(x) = \frac{2x^2}{x^2-4}$	$x=2$, $x=-2$	$y=2$	None	No Asymptotes
$f(x) = \frac{2x^2-7x+4}{x^2-4x+4}$	$x=2$	$y=2$	None	No Asymptotes

Applications of Rational Functions

Modeling Real-World Phenomena

Rational functions are used to model situations where quantities approach a limiting value, such as costs, concentrations, or rates. Asymptotic behavior is especially important in these contexts.

Example 1: Pollution Removal Cost

• Function: $C(p) = \frac{10}{10-p}$, where $C(p)$ is the cost (in millions of dollars) to remove $p$ percent of pollutants.

• Table of Values:

p (%)	0	2	4	6	8	9	9.5	9.9
$C(p)$	1	1.25	1.67	2.5	5	10	20	100

• Graph: The graph shows a sharp increase as $p$ approaches 10, indicating a vertical asymptote at $p=10$.

• Interpretation: As the percentage of pollutants removed increases, the cost increases dramatically, approaching infinity as $p$ nears 10%.

Example 2: Average Cost of Laptop Production

• Function: $C(x) = \frac{2000 + 120x}{x}$, where $x$ is the number of laptops produced.

• Table of Values:

x	10	50	100	500	1000	5000	10000
$C(x)$	320	160	140	124	122	120.4	120.2

• Horizontal Asymptote: As $x$ becomes very large, $C(x)$ approaches $y=120$.

• Interpretation: The average cost per laptop approaches $120$ as more laptops are produced, reflecting economies of scale.

Practice Problems and Further Applications

Drug Concentration in Bloodstream

• Function: $C(t) = \frac{2}{t+2}$, where $C(t)$ is the concentration (mg/L) and $t$ is time (hours).

• Key Questions:

  • Find the concentration after a given time.

  • Determine how long since administration for a given concentration.

  • Graph the function and interpret its behavior.

• Horizontal Asymptote: $y=0$ as $t \to \infty$.

Mixing Solutions: Glucose Example

• Function: $C(x) = \frac{2x}{x+2}$, where $x$ is the amount of glucose added (g).

• Table of Values:

x (g)	0	2	4	8	10	20	50
$C(x)$	0	1	1.33	1.6	1.67	1.82	1.96

• Horizontal Asymptote: $y=2$ as $x \to \infty$.

• Interpretation: The concentration approaches 2 mg/L as more glucose is added.

Summary of Key Concepts

• Vertical Asymptotes occur where the denominator of a rational function is zero and the numerator is nonzero.

• Horizontal Asymptotes describe the end behavior of the function as $x$ approaches infinity or negative infinity.

• Slant (Oblique) Asymptotes may occur when the degree of the numerator is exactly one more than the degree of the denominator.

• Applications include modeling costs, concentrations, and other phenomena where values approach a limit.

Example: In cost modeling, rational functions can show how costs increase rapidly as a target is approached, or how average costs decrease and level off as production increases.

Additional info: These notes cover the analysis and application of rational functions, focusing on asymptotic behavior and real-world modeling, which are central topics in College Algebra.