Graph the cational function. \[ f(x)=\frac{x^{2}-6 x+9}{x^{3}-7 x^{2}} \] Start by drawing the vertical and horizontal asxmptotes. Then plot the intercepts (ir ony), and plot at least one point on each side of each vertical asymptote. Finally, dick on the graph a-function button.

To graph the function \( f(x)=\frac{x^{2}-6 x+9}{x^{3}-7 x^{2}} \), we first need to find its vertical and horizontal asymptotes. Vertical asymptotes occur where the denominator equals zero, which means we need to solve \( x^3 - 7x^2 = 0 \). Factoring gives us \( x^2(x - 7) = 0 \), leading to vertical asymptotes at \( x = 0 \) and \( x = 7 \). For horizontal asymptotes, we look at the degrees of the polynomial. The degree of the numerator is 2 and the degree of the denominator is 3. Since the degree of the numerator is less than that of the denominator, there is a horizontal asymptote at \( y = 0 \). Next, let’s find the intercepts. For the x-intercept, set the numerator equal to zero: \( x^2 - 6x + 9 = 0 \) simplifies to \( (x - 3)^2 = 0 \), giving an x-intercept at \( x = 3 \). For the y-intercept, set \( x = 0 \): \( f(0) = \frac{0^2 - 6(0) + 9}{0^{3}-7(0^{2})} \), which is technically undefined due to a zero denominator. Finally, we can select points on either side of the vertical asymptotes (for example, \( x = -1 \) and \( x = 1 \) for the left side of \( x = 0 \); \( x = 5 \) and \( x = 8 \) for the right of \( x = 7 \)). Evaluating these points gives you a sense of the function's behavior. Remember, when plotting, be crisp with those asymptotes! They definitely serve as boundaries for your function's sketch. Try using a graphing tool or calculator to accurately visualize this fascinating rational function!