Describe the left-hand and right-hand behavior of the graph of the polynomial function. (Select all that apply.) h(x) = 4 − 0.6x5th power − 2.9x3 cubed

To determine the left-hand and right-hand behavior of the graph of the polynomial function \( h(x) = 4 - 0.6x^5 - 2.9x^3 \), we need to analyze the highest degree term, as it dictates the end behavior of the polynomial. Here’s the step-by-step breakdown:

1. **Degree of the Polynomial**: The highest power of \( x \) is \( x^5 \), which is an odd degree (degree 5).
   
2. **Leading Coefficient**: The leading term is \( -0.6x^5 \), where the coefficient is negative (-0.6).

### Left-Hand Behavior (as \( x \to -\infty \)):
Since the leading term is \( -0.6x^5 \) (odd degree and negative leading coefficient), as \( x \to -\infty \), the term \( -0.6x^5 \to +\infty \). Therefore:
- As \( x \to -\infty \), \( h(x) \to +\infty \).

### Right-Hand Behavior (as \( x \to +\infty \)):
For large positive \( x \), the leading term \( -0.6x^5 \) dominates and behaves as \( -0.6x^5 \to -\infty \). Therefore:
- As \( x \to +\infty \), \( h(x) \to -\infty \).

### Summary of the End Behavior:
- **Left-hand behavior**: As \( x \to -\infty \), \( h(x) \to +\infty \).
- **Right-hand behavior**: As \( x \to +\infty \), \( h(x) \to -\infty \).

Would you like more details or have any further questions?

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Here are 5 questions to expand on this:

1. How does the degree of a polynomial affect its graph's end behavior?
2. What is the impact of a negative leading coefficient on the behavior of a polynomial function?
3. How can you determine the turning points of a polynomial function like \( h(x) \)?
4. What happens to the graph near the roots of a cubic term compared to a quintic term?
5. How can you graph a polynomial function manually based on its equation?

**Tip**: The leading term of a polynomial (the term with the highest degree) dominates the end behavior of the function as \( x \to \pm\infty \).

Describe the left-hand and right-hand behavior of the graph of the polynomial function. (Select all that apply.) h(x) = 4 − 0.6x^5 − 2.9x^3

Understand the left-hand and right-hand behavior of the polynomial function h(x) = 4 − 0.6x^5 − 2.9x^3. This problem discusses key polynomial concepts such as degree, leading coefficient, and their influence on the graph's end behavior. Suitable for high school students in Grades 9-12.

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