Graph the rational function. $$ f(x)=\frac{-16}{x^{2}-2 x-8} $$ Start by drawing the vertical and horizontal asymptotes. Then plot the intercepts (if any), and plot at least one point on each side of each vertical asymptote. Finally, click on the graph-a-function button.

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To graph the rational function

$$
f(x) = \frac{-16}{x^2 - 2x - 8},
$$

we will follow the steps outlined: finding vertical and horizontal asymptotes, intercepts, and additional points.

### Step 1: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero (and the numerator is not zero).

First, we need to factor the denominator:

$$
x^2 - 2x - 8 = (x - 4)(x + 2).
$$

Setting the denominator equal to zero gives us:

$$
(x - 4)(x + 2) = 0.
$$

Thus, the vertical asymptotes are at:

$$
x = 4 \quad \text{and} \quad x = -2.
$$

### Step 2: Find Horizontal Asymptotes

To find the horizontal asymptote, we look at the degrees of the numerator and denominator. The degree of the numerator is 0 (since it is a constant -16), and the degree of the denominator is 2.

Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is:

$$
y = 0.
$$

### Step 3: Find Intercepts

**Y-Intercept:**
To find the y-intercept, we evaluate $$ f(0) $$:

$$
f(0) = \frac{-16}{0^2 - 2(0) - 8} = \frac{-16}{-8} = 2.
$$

So, the y-intercept is at $$ (0, 2) $$.

**X-Intercept:**
To find the x-intercept, we set the numerator equal to zero. However, since the numerator is a constant (-16), it never equals zero. Therefore, there are no x-intercepts.

### Step 4: Plot Points Around Vertical Asymptotes

We will choose points to the left and right of each vertical asymptote.

**For $$ x = -2 $$:**
- Choose $$ x = -3 $$:
  $$
  f(-3) = \frac{-16}{(-3)^2 - 2(-3) - 8} = \frac{-16}{9 + 6 - 8} = \frac{-16}{7} \approx -2.29.
  $$
- Choose $$ x = -1 $$:
  $$
  f(-1) = \frac{-16}{(-1)^2 - 2(-1) - 8} = \frac{-16}{1 + 2 - 8} = \frac{-16}{-5} = 3.2.
  $$

**For $$ x = 4 $$:**
- Choose $$ x = 3 $$:
  $$
  f(3) = \frac{-16}{(3)^2 - 2(3) - 8} = \frac{-16}{9 - 6 - 8} = \frac{-16}{-5} = 3.2.
  $$
- Choose $$ x = 5 $$:
  $$
  f(5) = \frac{-16}{(5)^2 - 2(5) - 8} = \frac{-16}{25 - 10 - 8} = \frac{-16}{7} \approx -2.29.
  $$

### Summary of Points
- Vertical Asymptotes: $$ x = -2 $$ and $$ x = 4 $$
- Horizontal Asymptote: $$ y = 0 $$
- Y-Intercept: $$ (0, 2) $$
- Points:
  - $$ (-3, -2.29) $$
  - $$ (-1, 3.2) $$
  - $$ (3, 3.2) $$
  - $$ (5, -2.29) $$

### Step 5: Graph the Function

Now, you can plot the vertical and horizontal asymptotes, the intercepts, and the points calculated above. The graph will approach the asymptotes but will not touch them.

After plotting all the points and asymptotes, you can visualize the behavior of the function around the vertical asymptotes and the horizontal asymptote.

If you have graphing software or a graphing calculator, you can input the function directly to see the complete graph.
To graph $$ f(x) = \frac{-16}{x^2 - 2x - 8} $$, identify vertical asymptotes at $$ x = -2 $$ and $$ x = 4 $$, horizontal asymptote at $$ y = 0 $$, y-intercept at $$ (0, 2) $$, and no x-intercepts. Plot points near the asymptotes to visualize the function's behavior.

Graph the rational function. \[ f(x)=\frac{-16}{x^{2}-2 x-8} \] Start by drawing the vertical and horizontal asymptotes. Then plot the intercepts (if any), and plot at least one point on each side of each vertical asymptote. Finally, click on the graph-a-function button.

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