17. Graph the rational function $$ r(x)=\frac{x-9}{x^{2}-4} $$

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1. **Factor the denominator**

The denominator factors as  
   $$
   x^{2} - 4 = (x-2)(x+2).
   $$

2. **Determine the domain**

The function is undefined when the denominator is zero. Set  
   $$
   x^{2} - 4 = 0 \quad \Rightarrow \quad x = 2 \text{ or } x = -2.
   $$
   Thus, the domain is  
   $$
   (-\infty,-2) \cup (-2,2) \cup (2,\infty).
   $$

3. **Find vertical asymptotes**

Since there are no common factors between the numerator $$ x-9 $$ and the factors of the denominator, the zeros of the denominator yield vertical asymptotes. Therefore, the vertical asymptotes are at  
   $$
   x = 2 \quad \text{and} \quad x = -2.
   $$

4. **Find the horizontal asymptote**

The degree of the numerator is 1 and the degree of the denominator is 2. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is  
   $$
   y = 0.
   $$

5. **Find the intercepts**

- **x-intercept:** Set the numerator equal to zero:
     $$
     x-9=0 \quad \Rightarrow \quad x=9.
     $$
     The x-intercept is at $$(9, 0)$$, provided $$ 9 $$ is in the domain (which it is).
     
   - **y-intercept:** Evaluate $$ r(x) $$ at $$ x = 0 $$:
     $$
     r(0)=\frac{0-9}{0^{2}-4}=\frac{-9}{-4}=\frac{9}{4}.
     $$
     The y-intercept is at $$\left(0, \frac{9}{4}\right)$$.

6. **Analyze the end behavior and sketch the graph**

- **End behavior:**  
     As $$ x \to \pm\infty $$,
     $$
     r(x)=\frac{x-9}{x^2-4} \approx \frac{x}{x^2}=\frac{1}{x}\to 0,
     $$
     which is consistent with the horizontal asymptote $$ y = 0 $$.

- **Behavior near the vertical asymptotes:**  
     Examine the limits as $$ x $$ approaches $$ 2 $$ and $$ -2 $$ from the left and right.  
     
     For $$ x \to 2^+ $$ and $$ x \to 2^- $$, the sign of each factor in $$ (x-2)(x+2) $$ will dictate whether the function goes to $$ +\infty $$ or $$ -\infty $$. A similar analysis applies for $$ x \to -2 $$.  
     
     (A detailed sign chart can be set up to determine the behavior on each interval by testing sample points.)

7. **Sketching the graph**

- Plot the vertical asymptotes $$ x = 2 $$ and $$ x = -2 $$ as dashed lines.
   - Draw the horizontal asymptote $$ y = 0 $$ as a dashed horizontal line.
   - Plot the x-intercept at $$(9,0)$$ and the y-intercept at $$\left(0,\frac{9}{4}\right)$$.
   - Use the end behavior as $$ x \to \pm\infty $$ approaching $$ y = 0 $$ and the behavior near vertical asymptotes to sketch the branches of the graph in each interval.

The final graph of  
$$
r(x)=\frac{x-9}{x^{2}-4}
$$
will show two vertical asymptotes at $$ x = -2 $$ and $$ x = 2 $$, a horizontal asymptote at $$ y = 0 $$, and intercepts at $$(9,0)$$ and $$\left(0,\frac{9}{4}\right)$$.
To graph the rational function $$ r(x) = \frac{x-9}{x^{2}-4} $$, follow these steps: 1. **Factor the denominator**: $$ x^{2} - 4 = (x-2)(x+2) $$. 2. **Determine the domain**: Exclude $$ x = 2 $$ and $$ x = -2 $$. 3. **Find vertical asymptotes**: At $$ x = 2 $$ and $$ x = -2 $$. 4. **Find the horizontal asymptote**: $$ y = 0 $$. 5. **Find intercepts**: - x-intercept at $$ x = 9 $$. - y-intercept at $$ \left(0, \frac{9}{4}\right) $$. 6. **Analyze end behavior**: As $$ x \to \pm\infty $$, $$ r(x) \to 0 $$. 7. **Sketch the graph**: Draw vertical asymptotes at $$ x = 2 $$ and $$ x = -2 $$, horizontal asymptote at $$ y = 0 $$, and plot the intercepts. Sketch the branches based on the asymptotes and intercepts. The graph will have two vertical asymptotes, a horizontal asymptote, and intercepts at specified points.

Graph the rational function

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