Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Polynomial Function Analysis
**Question 7 [−/1 Points]**
**Objective:** Use the graph to write the formula for a polynomial function of least degree.
**Function:** \( f(x) = \_\_\_\_\_\_ \)
#### Graph Description:
- The graph shows a polynomial curve that resembles a cubic function.
- It crosses the x-axis at approximately \(-2\), \(0\), and \(2\).
- The graph shows two turning points:
- A peak above the x-axis near \(x = -1\).
- A trough below the x-axis near \(x = 1\).
- The function appears to rise towards positive infinity as \(x\) increases and decrease towards negative infinity as \(x\) decreases, suggesting an odd-degree polynomial.
#### Explanation:
To determine the polynomial of least degree that fits this graph, observe the x-intercepts and turning points:
- X-intercepts: \(-2\), \(0\), \(2\) may correspond to the roots, implying factors of \((x + 2)\), \(x\), and \((x - 2)\).
- The simplest polynomial resembling this behavior is likely of third degree due to the three roots and two turning points.
**Additional Materials:**
- [eBook]
- [Example Video]
**Note:** Consider checking your work by verifying if the polynomial function you derive matches the graph's features.
#### Conclusion:
Use the intercepts and features to construct a basic cubic polynomial, possibly in the form:
\[ f(x) = a(x + 2)(x)(x - 2) \]
where \(a\) is a scalar adjusted to match the specific behavior and scaling observed on the graph.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9150cb0-c0bc-425f-af9e-d612c3f65210%2F1401722d-4a8f-4c04-832b-d69b286bcf52%2F4rruq7_processed.png&w=3840&q=75)
Transcribed Image Text:### Polynomial Function Analysis
**Question 7 [−/1 Points]**
**Objective:** Use the graph to write the formula for a polynomial function of least degree.
**Function:** \( f(x) = \_\_\_\_\_\_ \)
#### Graph Description:
- The graph shows a polynomial curve that resembles a cubic function.
- It crosses the x-axis at approximately \(-2\), \(0\), and \(2\).
- The graph shows two turning points:
- A peak above the x-axis near \(x = -1\).
- A trough below the x-axis near \(x = 1\).
- The function appears to rise towards positive infinity as \(x\) increases and decrease towards negative infinity as \(x\) decreases, suggesting an odd-degree polynomial.
#### Explanation:
To determine the polynomial of least degree that fits this graph, observe the x-intercepts and turning points:
- X-intercepts: \(-2\), \(0\), \(2\) may correspond to the roots, implying factors of \((x + 2)\), \(x\), and \((x - 2)\).
- The simplest polynomial resembling this behavior is likely of third degree due to the three roots and two turning points.
**Additional Materials:**
- [eBook]
- [Example Video]
**Note:** Consider checking your work by verifying if the polynomial function you derive matches the graph's features.
#### Conclusion:
Use the intercepts and features to construct a basic cubic polynomial, possibly in the form:
\[ f(x) = a(x + 2)(x)(x - 2) \]
where \(a\) is a scalar adjusted to match the specific behavior and scaling observed on the graph.
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