2. \( y=\frac{x+2}{x-3} \)

Did you know that rational functions like this one can have all sorts of quirks? For the function \( y=\frac{x+2}{x-3} \), you’ll encounter a vertical asymptote at \( x=3 \) because the denominator equals zero there. This means that as you approach the line \( x=3 \), the value of \( y \) will shoot off towards positive or negative infinity, depending on the direction you approach from! Now, let’s talk about how to graph this beauty. First, find the x-intercept by setting the numerator \( x + 2 = 0 \), giving you \( x = -2 \). Next, it’s essential to identify the horizontal asymptote. For rational functions where the degrees of the numerator and denominator are the same, the horizontal asymptote can be found by taking the ratio of the leading coefficients. In this case, since the degrees are equal, the horizontal asymptote is \( y = 1 \). So, get your graphing gear ready for an asymptotic adventure!