Asymptotes are important for understanding how certain math functions, called rational functions, act, especially when they meet the x-axis (the horizontal line) and the y-axis (the vertical line). This is really helpful for Year 12 students who are trying to learn about graphs of functions.

Types of Asymptotes

1. Vertical Asymptotes: These happen when the function can’t be defined, which usually occurs when the bottom part of the fraction equals zero. For example, in the function

   $f(x) = \frac{1}{x-3}$

   there is a vertical asymptote at $x = 3$. This means that as we get close to $x = 3$, the values of $f(x)$ go up or down really fast, making it like a wall that the graph can’t pass.

2. Horizontal Asymptotes: These show us what happens to the function as $x$ gets really big (positive or negative). For example, in this rational function:

   $g(x) = \frac{2x + 3}{x + 4}$

   as $x$ goes to really high or low numbers, this function gets closer to a horizontal line at $y = 2$. This is based on the main parts of the top and bottom of the fraction.

Intersection Points and Asymptotes

• X-Intercepts: A rational function meets the x-axis where $f(x) = 0$. This happens when the top part, or the numerator, equals zero. For example, in the function

  $h(x) = \frac{x^2 - 4}{x - 1}$

  the x-intercepts are at $x = -2$ and $x = 2$. If these points are the same as where a vertical asymptote is, the function won't meet the x-axis there.

• Y-Intercepts: To find the y-intercept, we check what $f(0)$ equals. Asymptotes can change where we find the y-intercept. If there's a vertical asymptote before $x = 0$, we won’t have a y-intercept there. But, if the vertical asymptote is after $x = 0$, we might still find a y-intercept.

Summary

In short, vertical and horizontal asymptotes help us understand how graphs behave and where they might meet the axes. They show us boundaries where the function can’t go and help us guess what it will look like as $x$ gets closer to certain numbers. Always remember: a vertical asymptote means the function can’t cross that line, while horizontal asymptotes help us see the function's long-term behavior on the graph.