Consider the polynomial function 
\[h(x)=3x^7+x^4+5x^3-80\].
What is the end behavior of the graph of 
\[h\ \]?

To determine the end behavior of the polynomial function
\[
h(x) = 3x^7 + x^4 + 5x^3 - 80,
\]
we need to focus on the term with the highest degree, as this term dominates the behavior of the polynomial as \(x\) becomes very large or very small.

### Step-by-step analysis:

1. **Identify the degree**:  
   The highest degree term in the polynomial is \(3x^7\), so the degree of the polynomial is 7.

2. **Leading coefficient**:  
   The leading coefficient (the coefficient of the highest degree term) is 3.

3. **Degree and leading coefficient implications**:
   - Since the degree is odd (7), the end behavior will differ for large positive and large negative values of \(x\).
   - Since the leading coefficient is positive (3), the graph will rise to infinity as \(x\) approaches \(+\infty\), and fall to negative infinity as \(x\) approaches \(-\infty\).

### End behavior:
- As \(x \to +\infty\), \(h(x) \to +\infty\).
- As \(x \to -\infty\), \(h(x) \to -\infty\).

Thus, the graph of \(h(x)\) behaves as follows:
- **Rises** to the right (as \(x \to +\infty\)).
- **Falls** to the left (as \(x \to -\infty\)).

### Summary:
The end behavior of the function \(h(x) = 3x^7 + x^4 + 5x^3 - 80\) is:
\[
\lim_{x \to +\infty} h(x) = +\infty \quad \text{and} \quad \lim_{x \to -\infty} h(x) = -\infty.
\]

Do you want more details or have any questions?

Here are some related questions to deepen understanding:
1. How does the leading term \(3x^7\) impact the overall shape of the graph?
2. What happens to the other terms (\(x^4\), \(5x^3\)) as \(x \to \infty\) or \(x \to -\infty\)?
3. How does the constant term \(-80\) affect the graph of the polynomial?
4. What would the end behavior be if the leading coefficient of \(x^7\) were negative?
5. How do even-degree polynomials differ in end behavior compared to odd-degree polynomials?

**Tip**: When analyzing polynomials, the term with the highest degree always dictates the end behavior, regardless of the lower-degree terms.

Consider the polynomial function h(x) = 3x^7 + x^4 + 5x^3 - 80. What is the end behavior of the graph of h?

Analyze the end behavior of the polynomial function h(x) = 3x^7 + x^4 + 5x^3 - 80. Understand how the leading term determines the behavior as x approaches infinity or negative infinity. Suitable for students in Grades 10-12, covering polynomial functions and leading term analysis.

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