Understanding how changes in the top and bottom parts of a fraction affect the lines we call asymptotes in rational functions can be quite enlightening. Let’s simplify this!

Vertical Asymptotes

Vertical asymptotes happen when the bottom part (denominator) of a fraction equals zero, and the top part (numerator) does NOT equal zero.

For example, consider this function:

$$f(x) = \frac{p(x)}{q(x)}$$

Here, $p(x)$ is the top part and $q(x)$ is the bottom part.

If we change the bottom part to something like $q(x) = x - 2$, we will have a vertical asymptote at $x = 2$.

Changing $q(x)$ affects where these asymptotes are located. For example, if we change the bottom part to $q(x) = (x - 2)(x + 1)$, we end up with vertical asymptotes at both $x = 2$ and $x = -1$.

Horizontal Asymptotes

Horizontal asymptotes are more about the degrees (the highest power of x) of the top and bottom parts. These are key when looking at how the function acts at very high or very low values of x.

• If the degree of the top part is less than the degree of the bottom part: The horizontal asymptote is at $y = 0$.
• If the degrees are the same: The horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients (the numbers in front of the highest power) of the top and bottom parts.
• If the top part's degree is greater: There’s no horizontal asymptote (the function goes off to infinity).

Summing It Up

In summary, changing the top or bottom part of a fraction not only moves the vertical asymptotes but can also change the horizontal ones based on their degrees. To really understand these changes, look at the degrees and where the zeros (points where the function equals zero) of your functions are. It’s all about understanding how the function behaves. Once you get the hang of it, this can really make sense of tricky rational functions!