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match the function with the correct graph of its end behavior model.
1.) y =2x^3-3x^2+1/x+3
2.)y = x^5-x^4+x+1/2x^2+x-3
3.) y = 2x^4-x^3+x^2-1/2-x
4.) y = x^4-3x^3+x^2-1/1-x^2

To match the given functions to their corresponding graphs based on the end behavior model, we need to look at the highest-degree terms of each polynomial. The end behavior of a function is primarily determined by the leading term (the term with the highest power of \( x \)).

Let's break down the end behaviors for each function:

1. **\( y = 2x^3 - 3x^2 + \frac{1}{x} + 3 \)**:
   - The leading term is \( 2x^3 \). 
   - The end behavior is that as \( x \to \infty \), \( y \to \infty \), and as \( x \to -\infty \), \( y \to -\infty \).
   - This corresponds to a cubic graph, which typically goes down to the left and up to the right.

2. **\( y = x^5 - x^4 + x + \frac{1}{2x^2} + x - 3 \)**:
   - The leading term is \( x^5 \).
   - As \( x \to \infty \), \( y \to \infty \), and as \( x \to -\infty \), \( y \to -\infty \).
   - This graph will resemble a quintic function, which usually rises on both ends but has more complex behavior in the middle.

3. **\( y = 2x^4 - x^3 + x^2 - \frac{1}{2} - x \)**:
   - The leading term is \( 2x^4 \).
   - As \( x \to \infty \), \( y \to \infty \), and as \( x \to -\infty \), \( y \to \infty \) (since the leading term is even).
   - The graph is expected to rise on both ends like a quartic function.

4. **\( y = x^4 - 3x^3 + x^2 - \frac{1}{1 - x^2} \)**:
   - The leading term is \( x^4 \).
   - Like the previous function, this also rises on both ends because \( x^4 \) dominates as \( x \to \infty \) or \( x \to -\infty \).
   - The graph should look similar to a quartic curve, but with some different behavior around the middle due to lower-degree terms.

### Matching to Graphs:
- **Graph (a)**: This graph shows a curve going down to the left and up to the right, consistent with a cubic function. Therefore, it matches **Function 1**.
- **Graph (b)**: Both ends rise upward, which matches the behavior of a quartic function. It corresponds to **Function 3 or 4**.
- **Graph (c)**: The graph resembles a quintic function with a positive leading coefficient because it goes down on the left and up on the right. This matches **Function 2**.
- **Graph (d)**: Both ends rise, similar to graph (b), which suggests a quartic function. Therefore, this would correspond to **Function 4 or 3**.

### Final Answer:
- **Graph (a)**: Function 1 (Cubic function).
- **Graph (b)**: Function 3 (Quartic function).
- **Graph (c)**: Function 2 (Quintic function).
- **Graph (d)**: Function 4 (Quartic function).

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How does the degree of a polynomial function affect its graph?
2. What are the differences in the end behavior between odd-degree and even-degree polynomials?
3. Can you describe how the lower-degree terms affect the graph near the origin?
4. How would the graph change if the leading coefficient were negative for these functions?
5. What role do vertical or horizontal asymptotes play in the behavior of these graphs?

**Tip:** When analyzing polynomial graphs, always start by focusing on the highest-degree term to determine the end behavior!

Match the function with the correct graph of its end behavior model:
1.) y = 2x^3 - 3x^2 + 1/x + 3
2.) y = x^5 - x^4 + x + 1/2x^2 + x - 3
3.) y = 2x^4 - x^3 + x^2 - 1/2 - x
4.) y = x^4 - 3x^3 + x^2 - 1/1 - x^2

Analyze polynomial functions based on their leading terms to determine the end behavior and match them with corresponding graphs. Functions include cubic, quartic, and quintic equations. Suitable for students in Grades 10-12.

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