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The given function is:

\[
f(x) = \frac{-6x^3 + 9x^2 + 177x - 90}{2x^2 + x - 55} = \frac{3(1 - 2x)(x - 6)(x + 5)}{(2x + 11)(x - 5)}
\]

### To find the end behavior model:
We are interested in what happens to the function as \( x \) approaches \( \pm \infty \). To do this, we focus on the leading terms in both the numerator and the denominator, since these terms will dominate the behavior of the function for large values of \( x \).

#### 1. Leading terms in the numerator:
The numerator is:
\[
-6x^3 + 9x^2 + 177x - 90
\]
The leading term (highest degree) is \( -6x^3 \).

#### 2. Leading terms in the denominator:
The denominator is:
\[
2x^2 + x - 55
\]
The leading term (highest degree) is \( 2x^2 \).

#### 3. End behavior model:
For large \( x \), we approximate the function using the leading terms:
\[
f(x) \sim \frac{-6x^3}{2x^2}
\]

Simplifying:
\[
f(x) \sim \frac{-6x^3}{2x^2} = -3x
\]

### Conclusion:
The end behavior model of the function \( f(x) \) is \( -3x \). This means as \( x \to \infty \), \( f(x) \to -\infty \), and as \( x \to -\infty \), \( f(x) \to \infty \).

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Would you like more details on this process, or do you have any other questions?

Here are some related questions:
1. How does factoring affect the interpretation of a rational function's behavior?
2. What is the significance of the degree of the numerator and denominator in determining the end behavior?
3. How do asymptotes relate to the end behavior of a function?
4. Can you explain how the behavior of \( f(x) \) would differ for smaller values of \( x \)?
5. What happens to \( f(x) \) near its vertical asymptotes?

**Tip:** When analyzing the end behavior, always prioritize the highest-degree terms in the polynomial because they dominate the function for large values of \( x \).

Find the end behavior model of the function f(x) = (-6x^3 + 9x^2 + 177x - 90) / (2x^2 + x - 55)

Learn how to find the end behavior model for the rational function f(x) = (-6x^3 + 9x^2 + 177x - 90) / (2x^2 + x - 55). This solution covers polynomial and rational function concepts, end behavior models, and algebraic simplifications, suitable for Grades 10-12.

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