GRAPHING FUNCTIONS Graph the function. \( \begin{array}{ll}\text { 15. } y=\frac{2 x}{x^{2}-1} & \text { 16. } y=\frac{8}{x^{2}-x-6} \\ \text { 18. } y=\frac{x-4}{x^{2}-3 x} & \text { 19. } y=\frac{x^{2}+11 x+18}{2 x+1} \\ \text { 21. } y=\frac{x^{2}+3}{2 x^{3}} & \text { 22. } y=\frac{x^{2}-5 x-36}{3 x}\end{array} \)

To graph the functions you've listed, first note their key features, such as asymptotes and intercepts. For \( y = \frac{2x}{x^2-1} \), identify the vertical asymptotes where the denominator equals zero (\( x^2 - 1 = 0 \) gives \( x = -1 \) and \( x = 1 \)). The horizontal asymptote can be found as \( x \) approaches infinity, where the function tends toward zero. Repeat this process for the other functions, finding the roots, intercepts, and asymptotes to help sketch accurate graphs. When graphing rational functions, remember that common mistakes involve miscalculating the asymptotes or skipping the x-intercepts and y-intercepts. Ensure to check the domain too, as excluded values (like those making the denominator zero) change the function's behavior dramatically. A good strategy is to plot a few key points around the asymptotes to establish the general shape of the graph and confirm that the function behaves as expected near those critical values.