Question
Refer to the polynomial function h(x) = 0 that you wrote in Question 1 of Finding the Zeros of a Polynomial Function. Describe and provide an explanation for the end behavior of h(x).
Answers
GPT-4o mini
For the polynomial function \( h(x) = 0 \), the polynomial is a constant function. Its characteristics and end behavior can be analyzed as follows:
### Description of End Behavior
1. **Constant Value:** The value of \( h(x) \) is always 0 for any value of \( x \). Therefore, regardless of how far \( x \) increases or decreases, \( h(x) \) remains constant at 0.
2. **End Behavior:** As \( x \) approaches positive infinity (\( x \to +\infty \)), \( h(x) \) remains 0.
3. **Negative Infinity:** Similarly, as \( x \) approaches negative infinity (\( x \to -\infty \)), \( h(x) \) again remains 0.
### Explanation
- **Graph Representation:** If you were to graph \( h(x) = 0 \), it would be a horizontal line that spans the width of the graph at the height of 0 on the y-axis.
- **Polynomial Characteristics:** Since \( h(x) \) is a polynomial of degree 0 (as it is constant), there are no leading terms that would influence the growth behavior of the polynomial as \( x \) approaches either infinity.
- **Behavior at Infinity:** Therefore, the end behavior of \( h(x) \) can be summarized as:
\[
\lim_{x \to +\infty} h(x) = 0 \quad \text{and} \quad \lim_{x \to -\infty} h(x) = 0
\]
In conclusion, the end behavior of the polynomial function \( h(x) = 0 \) is flat across all \( x \)—it remains at the constant value of 0 at both ends as \( x \) approaches positive and negative infinity.
### Description of End Behavior
1. **Constant Value:** The value of \( h(x) \) is always 0 for any value of \( x \). Therefore, regardless of how far \( x \) increases or decreases, \( h(x) \) remains constant at 0.
2. **End Behavior:** As \( x \) approaches positive infinity (\( x \to +\infty \)), \( h(x) \) remains 0.
3. **Negative Infinity:** Similarly, as \( x \) approaches negative infinity (\( x \to -\infty \)), \( h(x) \) again remains 0.
### Explanation
- **Graph Representation:** If you were to graph \( h(x) = 0 \), it would be a horizontal line that spans the width of the graph at the height of 0 on the y-axis.
- **Polynomial Characteristics:** Since \( h(x) \) is a polynomial of degree 0 (as it is constant), there are no leading terms that would influence the growth behavior of the polynomial as \( x \) approaches either infinity.
- **Behavior at Infinity:** Therefore, the end behavior of \( h(x) \) can be summarized as:
\[
\lim_{x \to +\infty} h(x) = 0 \quad \text{and} \quad \lim_{x \to -\infty} h(x) = 0
\]
In conclusion, the end behavior of the polynomial function \( h(x) = 0 \) is flat across all \( x \)—it remains at the constant value of 0 at both ends as \( x \) approaches positive and negative infinity.