https://file.aiquickdraw.com/mathbot/file/1729863137283cu36s4kh.jpg

The function in the image is \( f(x) = -2(x - 2)^2(x^2 - 25) \). Let's break down how to determine the end behavior, as part (a) requests.

### End Behavior Analysis:

1. **Degree of the Polynomial**: 
   - \( (x - 2)^2 \) has a degree of 2, and \( (x^2 - 25) \) can be factored as \( (x - 5)(x + 5) \), giving an additional degree of 2.
   - Therefore, the total degree of the polynomial is \( 2 + 2 = 4 \), making this a quartic function.

2. **Leading Term**: 
   - Since \( (x^2 - 25) = (x - 5)(x + 5) \), the expanded form of this polynomial would have an \( x^2 \) term.
   - The highest power of \( x \) from both parts is \( x^4 \), and multiplying by \( -2 \), the leading term is \( -2x^4 \).

3. **End Behavior**:
   - For large values of \( |x| \), the term \( -2x^4 \) dominates the behavior of the polynomial.
   - Since the leading term has a negative coefficient and an even degree, the end behavior is:
     - As \( x \to \infty \), \( f(x) \to -\infty \).
     - As \( x \to -\infty \), \( f(x) \to -\infty \).

### Conclusion:
- The graph of \( f(x) \) behaves like \( y = -2x^4 \) for large values of \( |x| \).

Would you like details on other parts or further explanation?

Here are 5 follow-up questions:
1. How do the roots of \( f(x) = 0 \) affect the graph's shape?
2. What role does the multiplicity of roots play in the graph?
3. How would you determine the local minima or maxima for this function?
4. How can we factor \( x^2 - 25 \) and why does it simplify graphing?
5. How do we identify turning points in a quartic function?

**Tip**: The end behavior of a polynomial is primarily determined by the degree and the leading coefficient, which simplifies predicting the graph’s behavior.

Graph the polynomial function f(x) = -2(x - 2)^2(x^2 - 25) using parts (a) through (e). Determine the end behavior of the graph of the function.

Explore how to determine the end behavior of the polynomial function f(x) = -2(x - 2)^2(x^2 - 25). Learn about key concepts such as leading terms, degree, and the effect of coefficients in predicting end behavior. Suitable for Pre-Calculus students.

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