vii. Sketch the graph of each rational function a. \( \quad f(x)=\frac{x+2}{x^{2}-9} \)

To sketch the graph of \( f(x) = \frac{x + 2}{x^{2} - 9} \), follow these steps: 1. **Domain:** All real numbers except \( x = 3 \) and \( x = -3 \). 2. **Intercepts:** - Y-intercept at \( \left(0, -\frac{2}{9}\right) \). - X-intercept at \( (-2, 0) \). 3. **Asymptotes:** - Vertical asymptotes at \( x = 3 \) and \( x = -3 \). - Horizontal asymptote at \( y = 0 \). 4. **Behavior:** - Approaches \( y = 0 \) as \( x \) approaches \( \pm\infty \). - Approaches \( \pm\infty \) near \( x = -3 \) and \( x = 3 \). 5. **Plot Points:** Use additional points to guide the curve. By plotting these features, you can accurately sketch the graph of the function.