Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of $$ y=f(x) $$. $$ f(x)=\frac{x^{2}+10 x+24}{x^{2}+8 x+16} $$ Find any horizontal asymptotes of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one horizontal asymptote, (Type an equation.) B. The function has two horizontal asymptotes. The top asymptote is $$ \square $$ and the bottom asymptote is (Type equations.) C. There are no horizontal asymptotes. Find any vertical asymptotes of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one vertical asymptote, (Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is $$ \square $$ and the rightmost asymptote is (Type equations.) C. There are no vertical asymptotes.

Question

UpStudy AI · Accepted Answer

To analyze the function $$ f(x) = \frac{x^{2} + 10x + 24}{x^{2} + 8x + 16} $$, we will follow these steps:

1. **Identify Horizontal Asymptotes:**
   - Horizontal asymptotes are determined by the degrees of the polynomial in the numerator and the denominator.
   - Both the numerator and denominator are quadratic polynomials (degree 2).
   - The horizontal asymptote can be found by taking the ratio of the leading coefficients.

2. **Identify Vertical Asymptotes:**
   - Vertical asymptotes occur where the denominator is equal to zero (and the numerator is not zero at those points).
   - We will find the roots of the denominator.

### Step 1: Find Horizontal Asymptotes

The leading coefficients of both the numerator and denominator are 1 (from $$ x^2 $$). Therefore, the horizontal asymptote is:

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

### Step 2: Find Vertical Asymptotes

To find the vertical asymptotes, we need to set the denominator equal to zero:

$$
x^{2} + 8x + 16 = 0
$$

This can be factored as:

$$
(x + 4)(x + 4) = 0
$$

Thus, we have a double root at $$ x = -4 $$. This means there is one vertical asymptote at:

$$
x = -4
$$

### Summary of Asymptotes

- **Horizontal Asymptote:** $$ y = 1 $$
- **Vertical Asymptote:** $$ x = -4 $$

### Final Choices

**Horizontal Asymptote:**
A. The function has one horizontal asymptote, $$ y = 1 $$.

**Vertical Asymptote:**
A. The function has one vertical asymptote, $$ x = -4 $$.

### Sketch of the Graph

To sketch the graph of $$ f(x) $$, we note the following:

- The horizontal asymptote at $$ y = 1 $$ indicates that as $$ x $$ approaches $$ \pm \infty $$, $$ f(x) $$ approaches 1.
- The vertical asymptote at $$ x = -4 $$ indicates that the function will approach $$ \pm \infty $$ as $$ x $$ approaches -4 from the left or right.

The graph will be continuous except at the vertical asymptote and will approach the horizontal asymptote as $$ x $$ moves away from the vertical asymptote.

Now, let's summarize the findings:

- **Horizontal Asymptote:** $$ y = 1 $$
- **Vertical Asymptote:** $$ x = -4 $$

This information can be used to sketch the graph of $$ f(x) $$.
- **Horizontal Asymptote:** $$ y = 1 $$ - **Vertical Asymptote:** $$ x = -4 $$

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Calculus Questions