Asymptote

What is an Asymptote?

Have you ever noticed a function graph that approaches a line but never touches it? Well, that line is called an asymptote!

An asymptote is a straight line that is approached by a function graph when its variable value approaches infinity or approaches a certain value. Imagine like you're walking towards a wall but never actually touching it, that's the concept of an asymptote.

Types of Asymptotes

There are three types of asymptotes you need to know:

Vertical Asymptote

A vertical asymptote is a vertical line that the graph approaches when the function value approaches positive or negative infinity.

Definition: The line $x = a$ is a vertical asymptote if:

• When $x$ approaches $a$ from the left, $f(x) \to \pm\infty$
• When $x$ approaches $a$ from the right, $f(x) \to \pm\infty$

How to find: For rational functions, vertical asymptotes occur when denominator = 0 (and numerator ≠ 0), or when $Q(x) = 0$ and $P(x) \neq 0$.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph approaches when $x$ approaches positive or negative infinity.

Definition: The line $y = b$ is a horizontal asymptote if:

• $\lim_{x \to \infty} f(x) = b$
• $\lim_{x \to -\infty} f(x) = b$

Oblique Asymptote (Oblique)

An oblique asymptote is a slanted line that the graph approaches when $x$ approaches infinity.

Definition: The line $y = mx + c$ is an oblique asymptote if:

Asymptotes in Rational Functions

Let's focus on rational functions $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials.

Finding Vertical Asymptotes

Steps:

1. Find the value of $x$ that makes $Q(x) = 0$
2. Check if $P(x) \neq 0$ at that value
3. If yes, then there is a vertical asymptote at $x = a$

Example: Determine the vertical asymptote of $f(x) = \frac{x + 3}{x - 2}$

Solution:

• Denominator is zero when: $x - 2 = 0$, so $x = 2$
• When $x = 2$, numerator = $2 + 3 = 5 \neq 0$
• Therefore, vertical asymptote: $x = 2$

Let's look at the function behavior around the vertical asymptote:

Finding Horizontal Asymptotes

Rules for rational functions:

Let the degree of numerator = $m$ and degree of denominator = $n$

1. If $m < n$: Horizontal asymptote is $y = 0$
2. If $m = n$: Horizontal asymptote is $y = \frac{a}{b}$ (ratio of leading coefficients)
3. If $m > n$: No horizontal asymptote (but there might be an oblique asymptote)

Example: Determine the horizontal asymptote of:

1. $f(x) = \frac{2x + 1}{x^2 - 4}$

   Solution:

   • Degree of numerator = 1, degree of denominator = 2
   • Since 1 < 2, horizontal asymptote: $y = 0$
2. $g(x) = \frac{3x^2 - 1}{2x^2 + 5}$

   Solution:

   • Degree of numerator = 2, degree of denominator = 2
   • Since degrees are equal, horizontal asymptote: $y = \frac{3}{2}$

Let's see how the function approaches the horizontal asymptote:

Finding Oblique Asymptotes

Oblique asymptotes appear when the degree of numerator = degree of denominator + 1.

How to find: Perform polynomial division.

Example: Determine the oblique asymptote of $f(x) = \frac{x^2 + 2x - 1}{x - 1}$

Solution: Using polynomial division:

When $x \to \pm\infty$, the term $\frac{2}{x - 1} \to 0$

Therefore, oblique asymptote: $y = x + 3$

Drawing Graphs with Asymptotes

Asymptotes are very helpful in drawing function graphs. Here are the steps:

1. Determine all asymptotes (vertical, horizontal, or oblique)
2. Draw asymptotes with dashed lines
3. Find intercepts with the axes
4. Determine some additional points
5. Draw the curve that approaches the asymptotes

Complete Example: Draw the graph of $f(x) = \frac{x + 1}{x - 2}$

Step 1: Find asymptotes

• Vertical asymptote: $x = 2$ (denominator = 0)
• Horizontal asymptote: $y = 1$ (same degree, coefficient ratio = 1/1)

Step 2: Intercepts

• y-axis: $f(0) = \frac{0 + 1}{0 - 2} = -\frac{1}{2}$
• x-axis: $0 = \frac{x + 1}{x - 2}$, so $x = -1$

Step 3: Behavior around asymptotes

• When $x \to 2^-$: $f(x) \to -\infty$
• When $x \to 2^+$: $f(x) \to +\infty$
• When $x \to \pm\infty$: $f(x) \to 1$

Step 4: Value table to help with drawing

Practice Problems

1. Determine all asymptotes of $f(x) = \frac{2x^2 - 3x + 1}{x - 3}$

2. Determine the asymptotes of $g(x) = \frac{x^2 - 4}{x^2 - 9}$

3. The average cost function of a product is $C(x) = \frac{500 + 3x}{x}$. Determine the minimum cost per unit that can be achieved.

4. Draw a sketch of the graph $h(x) = \frac{x}{x^2 - 1}$ complete with its asymptotes.

Answer Key

Answer 1:

• Degree of numerator (2) = degree of denominator (1) + 1
• There is an oblique asymptote. By division: $f(x) = 2x + 3 + \frac{10}{x - 3}$
• Vertical asymptote: $x = 3$
• Oblique asymptote: $y = 2x + 3$

Answer 2:

• Vertical asymptote: $x^2 - 9 = 0$, so $x = 3$ and $x = -3$
• But when $x = 2$, numerator = 0, so $x = 2$ is not an asymptote
• When $x = -2$, numerator = 0, so $x = -2$ is not an asymptote
• Horizontal asymptote: $y = 1$ (same degree, ratio = 1/1)

Answer 3:

When $x \to \infty$, $\frac{500}{x} \to 0$ So minimum cost per unit = 3

Answer 4:

• Vertical asymptotes: $x = 1$ and $x = -1$
• Horizontal asymptote: $y = 0$ (degree of numerator < degree of denominator)
• The graph has three separate parts due to two vertical asymptotes

Value table for $h(x) = \frac{x}{x^2 - 1}$: