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The given function is \( f(x) = (x - 4)(x + 1)^2 \). Let's analyze it step by step to describe the errors in the graph and how to correct it:

### 1. **End Behavior:**
   - Since the highest degree term comes from expanding the expression \( (x - 4)(x + 1)^2 \), this is a cubic function. A cubic function with a positive leading coefficient should have the following end behavior:
     - As \( x \to \infty \), \( f(x) \to \infty \) (the function goes up to the right).
     - As \( x \to -\infty \), \( f(x) \to -\infty \) (the function goes down to the left).
   - The graph provided has incorrect end behavior as both ends point upwards. This should be corrected to have one end up (right) and the other down (left).

### 2. **X-intercepts:**
   - To find the x-intercepts, we set \( f(x) = 0 \):
     - \( (x - 4)(x + 1)^2 = 0 \) gives us \( x = 4 \) (single root) and \( x = -1 \) (double root).
   - The graph should cross the x-axis at \( x = 4 \) and touch the x-axis at \( x = -1 \) (because it is a double root, the graph will just touch the axis there without crossing it). 
   - The current graph incorrectly shows intercepts that are not aligned with these values.

### Corrections:
- The end behavior should be corrected to reflect that it goes down to the left and up to the right.
- The x-intercepts should be corrected to show \( x = 4 \) as a crossing point and \( x = -1 \) as a point where the graph just touches the x-axis (double root).

### Answer Choices:
- **The end behavior is wrong. It should be \( \downarrow \uparrow \) (down to the left, up to the right).**
- **The x-intercepts are wrong. They should be at \( (4,0) \) and \( (-1,0) \).**

Do you want details on how to sketch the correct graph or explanations of specific points?

### Related Questions:
1. How do you determine the end behavior of a polynomial function?
2. What is the significance of a double root in a polynomial function graph?
3. How do you find the x-intercepts of a polynomial function?
4. How can you use factoring to analyze the behavior of polynomial functions?
5. What role do multiplicity and degree play in shaping the graph of a function?

**Tip:** Always check the degree and leading coefficient of a polynomial to understand its end behavior.

Describe the error in graphing f(x) for the function, then graph it correctly.
The function is f(x) = (x - 4)(x + 1)^2.

Learn how to identify and correct the graphing errors in the cubic function f(x) = (x - 4)(x + 1)^2. This problem explains the correct end behavior and the proper placement of x-intercepts based on the function's roots and their multiplicities. Suitable for students in Grades 10-12.

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