Graph all vertical and horizontal asymptotes of the rational functio \[ f(x)=\frac{-9 x-4}{-3 x-3} \]

To find the vertical asymptotes of the rational function \( f(x) = \frac{-9x - 4}{-3x - 3} \), we look for values of \( x \) that make the denominator equal to zero. Setting the denominator \(-3x - 3 = 0\), we find: \[ -3x = 3 \implies x = -1. \] So, there is a vertical asymptote at \( x = -1 \). Next, to find the horizontal asymptote, we look at the degrees of the polynomial in the numerator and the denominator. Both the numerator and denominator are of degree 1. When the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficients here are -9 (from the numerator) and -3 (from the denominator). Thus, the horizontal asymptote is: \[ y = \frac{-9}{-3} = 3. \] In summary, the vertical asymptote is at \( x = -1 \) and the horizontal asymptote is at \( y = 3 \). --- Vertical and horizontal asymptotes aren’t just mathematical curiosities; they help in sketching graphs and understanding function behavior! Once you know these asymptotes, you can draw the graph more accurately, ensuring the function approaches the lines defined by the asymptotes as \( x \) moves toward the asymptote values. Understanding these concepts also enhances problem-solving in calculus, particularly when dealing with limits at infinity or around the vertical asymptotes. If these asymptotes create indeterminate forms, you might be inclined to use techniques like L'Hôpital's Rule for further analysis.