 more detailed breakdown of each graph and function pair?

Let's go through each function and graph pair in detail to determine which function corresponds to which graph based on the end behavior.

### Functions and Their End Behaviors:

1. **\(f(x) = x^4 + \ldots\)**
   - **Degree**: 4 (even)
   - **Leading coefficient**: Positive
   - **End Behavior**: As \(x \to \pm\infty\), \(f(x) \to +\infty\).  
   - **Category**: 1 (Both ends go to \(+\infty\))

2. **\(f(x) = 4x - 3x^6 + 5\)**
   - **Degree**: 6 (even)
   - **Leading coefficient**: Negative
   - **End Behavior**: As \(x \to \pm\infty\), \(f(x) \to -\infty\).  
   - **Category**: 2 (Both ends go to \(-\infty\))

3. **\(f(x) = -2x^3 + 4x - 7\)**
   - **Degree**: 3 (odd)
   - **Leading coefficient**: Negative
   - **End Behavior**: As \(x \to -\infty\), \(f(x) \to +\infty\); as \(x \to \infty\), \(f(x) \to -\infty\).  
   - **Category**: 4 (Left end goes to \(+\infty\), right end goes to \(-\infty\))

4. **\(f(x) = 3(x - 2)^2(x + 1)^3\)**
   - **Degree**: 5 (odd)
   - **Leading coefficient**: Positive
   - **End Behavior**: As \(x \to -\infty\), \(f(x) \to -\infty\); as \(x \to \infty\), \(f(x) \to +\infty\).  
   - **Category**: 3 (Left end goes to \(-\infty\), right end goes to \(+\infty\))

5. **\(f(x) = 9 - x^2(x^2 + 4x + 4)\)**
   - **Degree**: 4 (even)
   - **Leading coefficient**: Negative (after expansion)
   - **End Behavior**: As \(x \to \pm\infty\), \(f(x) \to -\infty\).  
   - **Category**: 2 (Both ends go to \(-\infty\))

6. **\(f(x) = -x^2(x - 1)(x + 4)\)**
   - **Degree**: 4 (even)
   - **Leading coefficient**: Negative
   - **End Behavior**: As \(x \to \pm\infty\), \(f(x) \to -\infty\).  
   - **Category**: 2 (Both ends go to \(-\infty\))

### Matching Functions to Graphs:

Now, let’s match each function to a graph based on their end behaviors:

1. **Graphs with both ends going upward (Category 1)**:
   - **Function**: \(f(x) = x^4 + \ldots\)
   - This function has an even degree (4) and a positive leading coefficient. The graph shows both ends going upward.
   - **Match**: The graph that shows both ends going up is matched with \(f(x) = x^4 + \ldots\).

2. **Graphs with both ends going downward (Category 2)**:
   - **Functions**: 
     - \(f(x) = 4x - 3x^6 + 5\)
     - \(f(x) = 9 - x^2(x^2 + 4x + 4)\)
     - \(f(x) = -x^2(x - 1)(x + 4)\)
   - All three functions have even degrees (6, 4, and 4 respectively) and negative leading coefficients, resulting in both ends going downward.
   - **Match**: The graphs that show both ends going down should be matched with these three functions.

3. **Graph with left end down and right end up (Category 3)**:
   - **Function**: \(f(x) = 3(x - 2)^2(x + 1)^3\)
   - This function has an odd degree (5) and a positive leading coefficient. The graph should show the left end going downward and the right end going upward.
   - **Match**: The graph with the left end going down and the right end going up is matched with \(f(x) = 3(x - 2)^2(x + 1)^3\).

4. **Graph with left end up and right end down (Category 4)**:
   - **Function**: \(f(x) = -2x^3 + 4x - 7\)
   - This function has an odd degree (3) and a negative leading coefficient. The graph should show the left end going upward and the right end going downward.
   - **Match**: The graph with the left end going up and the right end going down is matched with \(f(x) = -2x^3 + 4x - 7\).

### Detailed Matching:

- **Category 1**: Graph with both ends going up  
  - **Function**: \(f(x) = x^4 + \ldots\)

- **Category 2**: Graphs with both ends going down  
  - **Functions**:  
    - \(f(x) = 4x - 3x^6 + 5\)  
    - \(f(x) = 9 - x^2(x^2 + 4x + 4)\)  
   

more detailed breakdown of each graph and function pair?

This solution provides a step-by-step breakdown of how to match polynomial functions to their graphs based on end behaviors. It covers key concepts such as polynomial degrees, leading coefficients, and end behavior patterns for high school students in Grades 10-12.

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