https://file.aiquickdraw.com/mathbot/file/1726244359413qksn0pc4.jpg

The rational function given is:

\[
f(x) = \frac{-9(x - 5)(x^2 + 7)}{4(x + 2)(x - 8)}
\]

We are tasked with determining the end behavior of \( f(x) \) as \( x \to -\infty \) and \( x \to +\infty \).

### Step 1: Analyzing the degree of the polynomial
- **Numerator**: The highest degree term in the numerator comes from \( (x^2 + 7) \) and \( (x - 5) \). When expanded, the term with the highest degree will be \( x^3 \). So, the degree of the numerator is 3.
- **Denominator**: The highest degree term in the denominator comes from \( (x + 2)(x - 8) \). When expanded, the highest degree term is \( x^2 \). So, the degree of the denominator is 2.

### Step 2: Determining the end behavior
- As \( x \to \pm\infty \), the lower degree terms (constants like 5, 7, 2, and 8) become insignificant, and the function's behavior is dominated by the highest degree terms.
  
  Thus, we approximate:
  \[
  f(x) \sim \frac{-9x^3}{4x^2} = -\frac{9}{4}x
  \]
  - As \( x \to +\infty \), \( f(x) \to -\infty \).
  - As \( x \to -\infty \), \( f(x) \to +\infty \).

### Step 3: Choosing the correct answer
Based on this analysis:
- As \( x \to -\infty \), \( f(x) \to +\infty \).
- As \( x \to +\infty \), \( f(x) \to -\infty \).

This corresponds to the third option in the image:
- **"As x approaches negative infinity, the values of f(x) approach positive infinity, and as x approaches positive infinity, the values of f(x) approach negative infinity."**

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Would you like further details on how the end behavior is determined? Here are some related questions to think about:

1. How do the signs of the leading coefficients affect the end behavior of rational functions?
2. How does the degree of the numerator and denominator influence horizontal asymptotes?
3. What is the difference in end behavior when the degree of the numerator is higher than the denominator?
4. Can rational functions have slant (oblique) asymptotes? How do we find them?
5. How do vertical asymptotes and holes affect the graph of a rational function?

**Tip**: When determining the end behavior of rational functions, always compare the degrees of the numerator and denominator first.

Given the rational function f(x) = (-9(x - 5)(x^2 + 7)) / (4(x + 2)(x - 8)), which of the following is a true statement about the end behavior of f(x)?

Analyze the end behavior of the rational function f(x) = (-9(x - 5)(x^2 + 7)) / (4(x + 2)(x - 8)) by comparing the degrees of the numerator and denominator. As x approaches negative and positive infinity, find how the function behaves. Suitable for high school students in Grades 10-12.

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