Finding Horizontal and Vertical Asymptotes - 01-09-2014

Welcome back to Professor Baker's Math Class! Today, we revisited the crucial concepts of finding horizontal and vertical asymptotes. Understanding asymptotes is key to accurately graphing rational functions and comprehending their behavior.

Class Notes

Here's a review of what we covered in class. Remember, practice makes perfect, so don't hesitate to work through the examples and homework problems!

Vertical Asymptotes

Vertical asymptotes occur where the function is undefined, typically where the denominator of a rational function equals zero. Here's the process:

  1. Find Domain Restrictions: Determine the values of $x$ that make the denominator, $D(x)$, equal to zero. These are potential locations for vertical asymptotes. Mathematically, we're looking for values where $D(x) = 0$.
  2. Look for Common Factors: Simplify the rational function by canceling any common factors in the numerator and denominator. If a factor cancels, it creates a "hole" in the graph at that $x$ value instead of a vertical asymptote.
  3. Remaining Restrictions: All leftover domain restrictions after simplification represent vertical asymptotes.

For example, consider the function $$f(x) = \frac{x+2}{x^2 + 2x}$$.

First, factor the denominator: $$f(x) = \frac{x+2}{x(x+2)}$$.

Next, cancel the common factor of $(x+2)$: $$f(x) = \frac{1}{x}$$.

The original denominator, $x(x+2)$, equals zero when $x = 0$ and $x = -2$. However, since $(x+2)$ canceled, there's a hole at $x = -2$. The remaining restriction, $x = 0$, indicates a vertical asymptote at $x = 0$.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as $x$ approaches positive or negative infinity. To find them, compare the degrees of the numerator, $N(x)$, and the denominator, $D(x)$:

  1. Degree of Numerator vs. Denominator: Determine the highest power of $x$ in both the numerator and denominator. Let 'n' be the degree of $N(x)$ and 'd' be the degree of $D(x)$.
  2. Rules for Horizontal Asymptotes:
    • If $n < d$: The horizontal asymptote is $y = 0$.
    • If $n > d$: There is no horizontal asymptote.
    • If $n = d$: The horizontal asymptote is $y = \frac{\text{leading coefficient of } N(x)}{\text{leading coefficient of } D(x)}$.

Let's look at $$f(x) = \frac{2x^2}{x^2 - 1}$$.

Here, $n = 2$ and $d = 2$. Since $n = d$, the horizontal asymptote is $y = \frac{2}{1} = 2$.

Homework

For practice, complete problems 13-22 in the textbook. Focus on finding both the vertical and horizontal asymptotes for each function. Remember to show your work and think through each step! Good luck, and see you next time!