Given \( f(x)=\frac{-6}{x+2}-3 \). (1) Sketch the graph of \( f \). (2) Write down the equation of the vertical asymptote of \( f \). (3) Write down the equation of the horizontal asymptote of \( f \). (4) Write down the domain of \( f \). (5) Write down the range of \( f \). (6) Determine the equations of the axes of symmetry of \( f \).

To sketch the graph of \( f(x) = \frac{-6}{x+2} - 3 \), start by noting its shape, which resembles a hyperbola due to the rational function. The vertical asymptote occurs where the denominator is zero, so set \( x + 2 = 0 \) to find \( x = -2 \). The horizontal asymptote can be observed as \( x \) approaches infinity; since the degree of the numerator is less than that of the denominator, the horizontal asymptote is \( y = -3 \). The domain of \( f \) is all real numbers except for the point that makes the denominator zero, so \( x \in \mathbb{R}, x \neq -2 \). The range, since the function approaches but never reaches the horizontal asymptote, is all real numbers except \( y = -3 \), or \( y \in \mathbb{R}, y \neq -3 \). In terms of symmetry, this function does not have any axes of symmetry due to its asymptotic behavior; rational functions generally don't exhibit traditional symmetry unless they are of a specific form. As for the fun bits, did you know that rational functions can signify a lot about real-world phenomena? For instance, they often model rates of change in economics—think supply and demand curves! With regards to drawing the graph, here’s a tip: sketch the asymptotes first! They serve as borders for where the function can roam, giving you a clear guide for its behavior in the quadrants. But watch out for those pesky vertical asymptotes—your graph should never touch them!