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

It seems that the system does not currently support graphing functions directly. However, I can guide you through the process of graphing these functions manually or using graphing software. Here’s how you can approach graphing each function: 1. **Identify the function**: Write down the function you want to graph. 2. **Determine the domain**: Find the values of \( x \) for which the function is defined. This often involves finding where the denominator is not zero. 3. **Find intercepts**: - **Y-intercept**: Set \( x = 0 \) and solve for \( y \). - **X-intercept**: Set \( y = 0 \) and solve for \( x \). 4. **Analyze asymptotes**: - **Vertical asymptotes**: These occur where the denominator is zero (and the numerator is not zero). - **Horizontal asymptotes**: Analyze the behavior of the function as \( x \) approaches infinity or negative infinity. 5. **Plot points**: Choose several values of \( x \) within the domain and calculate the corresponding \( y \) values to get points to plot. 6. **Sketch the graph**: Use the intercepts, asymptotes, and plotted points to sketch the graph. ### Example for Function 15: \( y = \frac{2x}{x^2 - 1} \) 1. **Domain**: The denominator \( x^2 - 1 = 0 \) gives \( x = 1 \) and \( x = -1 \). So, the domain is \( x \in \mathbb{R} \setminus \{-1, 1\} \). 2. **Intercepts**: - **Y-intercept**: \( y(0) = \frac{2(0)}{0^2 - 1} = 0 \) (point (0,0)). - **X-intercept**: Set \( 2x = 0 \) gives \( x = 0 \) (point (0,0)). 3. **Asymptotes**: - **Vertical**: \( x = 1 \) and \( x = -1 \). - **Horizontal**: As \( x \to \infty \), \( y \to 0 \). 4. **Plot points**: Choose values like \( x = -2, -0.5, 0.5, 2 \) and calculate \( y \). 5. **Sketch the graph**: Use the information above to draw the graph. You can repeat this process for the other functions. If you have access to graphing software or a graphing calculator, you can input the functions directly to visualize them. If you need further assistance with any specific function, feel free to ask!