https://file.aiquickdraw.com/mathbot/file/1726617465965p36adw0x.jpg

Determine each feature of the graph of the given function.

Let's analyze the given function step by step. The function is:

\[
f(x) = \frac{2(x-3)}{5(x-3)(x+2)^2}
\]

### 1. **Horizontal Asymptotes:**
To determine horizontal asymptotes, we compare the degrees of the numerator and the denominator:

- Degree of the numerator: The numerator \( 2(x-3) \) is a linear term, so the degree is 1.
- Degree of the denominator: The denominator \( 5(x-3)(x+2)^2 \) contains a quadratic term \( (x+2)^2 \), so the degree is 3.

Since the degree of the numerator is smaller than the degree of the denominator, the horizontal asymptote is at \( y = 0 \).

**Conclusion**: The graph has a horizontal asymptote at \( y = 0 \).

### 2. **Vertical Asymptotes:**
Vertical asymptotes occur where the denominator equals zero, provided the numerator doesn't cancel the factor. The denominator is:

\[
5(x-3)(x+2)^2
\]

Setting the denominator equal to zero gives us:

\[
(x-3)(x+2)^2 = 0
\]

- \( x - 3 = 0 \) gives \( x = 3 \)
- \( (x+2)^2 = 0 \) gives \( x = -2 \)

Now, check for cancellations: the factor \( (x - 3) \) appears in both the numerator and denominator, so it cancels out. Therefore, \( x = 3 \) is a **hole**, not a vertical asymptote.

Thus, the only vertical asymptote is at \( x = -2 \).

**Conclusion**: There is 1 vertical asymptote at \( x = -2 \).

### 3. **Holes:**
Holes occur where a factor cancels from both the numerator and the denominator. In this case, the factor \( (x - 3) \) cancels, so there is a hole at \( x = 3 \).

**Conclusion**: There is a hole at \( x = 3 \).

### Summary:
- **Horizontal Asymptotes**: 1 (at \( y = 0 \))
- **Holes**: 1 (at \( x = 3 \))
- **Vertical Asymptotes**: 1 (at \( x = -2 \))

Let me know if you'd like further explanation or more details! Here are some related questions to expand on:

1. How do you determine the slant asymptote for a rational function?
2. What happens to the behavior of the function as \( x \to -2 \)?
3. Can there be more than one hole in a rational function?
4. How do you find the y-intercept of this function?
5. What is the domain of this function?

**Tip**: Always check for common factors between the numerator and denominator to identify holes in the graph.

Determine each feature of the graph of the given function f(x) = (2(x-3)) / (5(x-3)(x+2)^2)

Learn how to determine the features of the graph of a rational function, including horizontal asymptotes, vertical asymptotes, and holes. This example uses the function f(x) = (2(x-3)) / (5(x-3)(x+2)^2) to illustrate key concepts and formulas.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation