Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Show complete work on your worksheet! $$ f(x)=\frac{x^{2}-9}{x^{2}-x-12} $$ Reduced function: $$ f_{ ext {red }}(x)=\frac{x-3}{x-4} $$ Domain: x= VA: There is/are vertical asymptote(s) at Hole(s): There are no vertical asymptotes. O There is a hole at OThere is no hole in this function. O-intercept(s): OThere is/are $$ x $$-intercept(s) at There are no $$ x $$-intercepts.

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Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Show complete work on your worksheet! $$ f(x)=\frac{x^{2}-9}{x^{2}-x-12} $$ Reduced function: $$ f_{	ext {red }}(x)=\frac{x-3}{x-4} $$ Domain: x= VA: There is/are vertical asymptote(s) at Hole(s): There are no vertical asymptotes. O There is a hole at OThere is no hole in this function. O-intercept(s): OThere is/are $$ x $$-intercept(s) at There are no $$ x $$-intercepts.

UpStudy AI · Accepted Answer

Sure, let's analyze and sketch the graph of the rational function:

$$ f(x) = \frac{x^{2} - 9}{x^{2} - x - 12} $$

### Step 1: Factor and Simplify the Function

**Factor the numerator and the denominator:**

- **Numerator:** $$ x^{2} - 9 = (x - 3)(x + 3) $$
- **Denominator:** $$ x^{2} - x - 12 = (x - 4)(x + 3) $$

**Simplify the function by canceling the common factor $$(x + 3)$$:**

$$ f_{\text{red}}(x) = \frac{x - 3}{x - 4} $$

**Important:** The cancellation introduces a **hole** in the graph at $$ x = -3 $$ because the original function is undefined there.

### Step 2: Determine the Domain

The **domain** consists of all real numbers except where the denominator is zero.

- $$ x - 4 = 0 \Rightarrow x = 4 $$
- $$ x + 3 = 0 \Rightarrow x = -3 $$ (hole)

**Domain:** $$ x \neq 4 $$ and $$ x \neq -3 $$

### Step 3: Identify Vertical Asymptotes and Holes

- **Vertical Asymptote (VA):** Occurs where the denominator is zero but not canceled by the numerator.
  - $$ x = 4 $$

- **Hole(s):** Occurs where both the numerator and denominator are zero.
  - $$ x = -3 $$

**Summary:**
- **VA:** $$ x = 4 $$
- **Hole:** $$ x = -3 $$

### Step 4: Find Intercepts

**X-intercept(s):** Set the numerator of the reduced function to zero.

$$ x - 3 = 0 \Rightarrow x = 3 $$

- **X-intercept:** $$ (3, 0) $$

**Note:** $$ x = -3 $$ is **not** an x-intercept because it's a hole.

**Y-intercept:** Set $$ x = 0 $$ in the original function.

$$ f(0) = \frac{0^{2} - 9}{0^{2} - 0 - 12} = \frac{-9}{-12} = \frac{3}{4} $$

- **Y-intercept:** $$ \left(0, \frac{3}{4}\right) $$

### Step 5: Determine Horizontal Asymptote

Compare the degrees of the numerator and the denominator in the original function.

- Both are degree 2.

The horizontal asymptote is the ratio of the leading coefficients.

$$ y = \frac{1}{1} = 1 $$

- **Horizontal Asymptote:** $$ y = 1 $$

### Step 6: Sketch the Graph

1. **Plot the intercepts:**
   - X-intercept at $$ (3, 0) $$
   - Y-intercept at $$ \left(0, \frac{3}{4}\right) $$

2. **Draw the vertical asymptote:** A dashed line at $$ x = 4 $$.

3. **Mark the hole:** An open circle at $$ x = -3 $$.

4. **Draw the horizontal asymptote:** A dashed line at $$ y = 1 $$.

5. **Plot additional points if necessary** to accurately sketch the graph.

### Summary of Key Features

- **Reduced Function:** $$ f_{\text{red}}(x) = \frac{x - 3}{x - 4} $$
- **Domain:** $$ x \neq 4 $$ and $$ x \neq -3 $$
- **Vertical Asymptote:** $$ x = 4 $$
- **Hole:** $$ x = -3 $$
- **X-intercept:** $$ (3, 0) $$
- **Y-intercept:** $$ \left(0, \frac{3}{4}\right) $$
- **Horizontal Asymptote:** $$ y = 1 $$

### Graph of $$ f(x) $$

![Graph of the rational function](https://i.imgur.com/your-graph-link.png) *(Please plot the graph based on the above features)*

By following these steps and plotting the key features, you can accurately sketch the graph of the given rational function by hand.
To sketch the graph of $$ f(x) = \frac{x^{2} - 9}{x^{2} - x - 12} $$: 1. **Factor and Simplify:** - Numerator: $$ x^{2} - 9 = (x - 3)(x + 3) $$ - Denominator: $$ x^{2} - x - 12 = (x - 4)(x + 3) $$ - Simplified function: $$ f_{\text{red}}(x) = \frac{x - 3}{x - 4} $$ with a hole at $$ x = -3 $$ 2. **Domain:** $$ x \neq 4 $$ and $$ x \neq -3 $$ 3. **Intercepts:** - X-intercept: $$ (3, 0) $$ - Y-intercept: $$ \left(0, \frac{3}{4}\right) $$ 4. **Asymptotes:** - Vertical Asymptote: $$ x = 4 $$ - Horizontal Asymptote: $$ y = 1 $$ 5. **Hole:** At $$ x = -3 $$ 6. **Sketch the Graph:** - Plot the intercepts and asymptotes. - Draw the vertical asymptote at $$ x = 4 $$ and the horizontal asymptote at $$ y = 1 $$. - Mark a hole at $$ x = -3 $$. - Plot additional points to complete the graph. By following these steps, you can accurately sketch the graph of the rational function by hand.

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