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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![### Example Problem: Polynomial Equation from Graph
**Problem Statement:**
Determine an equation for the pictured graph. Write your answer in factored form and assume the leading coefficient is ±1.
**Graph Analysis:**
[Insert Graph Description]
The given graph shows a polynomial function intersecting the x-axis at multiple points. The graph appears to be a cubic polynomial with three real roots.
- The x-axis intersections (roots of the function) are approximately at x = -2, x = 1, and x = 3.
- The graph dips and rises, suggesting that it does not cross the x-axis in one smooth motion but has turning points indicating a polynomial with multiple terms.
**Graph Interpretation:**
The graph provided can be translated into a polynomial function in the following way:
1. Identify the roots of the function as seen on the x-axis crossings:
- Root 1: x = -2
- Root 2: x = 1
- Root 3: x = 3
2. Form the factors corresponding to each of these roots:
- For x = -2, the factor is (x + 2)
- For x = 1, the factor is (x - 1)
- For x = 3, the factor is (x - 3)
3. As the leading coefficient is either ±1, we can write the polynomial in factored form as:
\[
y = (x + 2)(x - 1)(x - 3)
\]
4. Verify the plotted graph's behavior, ensuring it matches the cubic nature with three intersections and corresponding changes in the slope.
**Answer Submission:**
`y = (x + 2)(x - 1)(x - 3)`
**Interactive Element:**
- [Submit Question Button]
- Allows students to verify their equation for correctness.
**Note:**
This exercise helps in understanding polynomial functions and how roots or zeros are identified and represented. The transformation of graphical behavior into a mathematical equation is a fundamental skill in algebra and calculus.
---
### Additional Resources:
- [Polynomial Functions: Concepts and Applications]
- [Graphing Techniques for Polynomial Equations]
- [Interactive Polynomial Graphing Tools]
### References:
- Polynomial graph interpretation and equation determination (Textbooks/Online Articles)
- Graphical tools used in educational settings (e.g., GeoGebra)
This example problem is designed for students](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0cf52f6a-9168-49cc-a8d7-0312a37994ea%2F7cf2d334-62b2-48c3-929a-2a4de30e8cb6%2Fzdz9q18_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Example Problem: Polynomial Equation from Graph
**Problem Statement:**
Determine an equation for the pictured graph. Write your answer in factored form and assume the leading coefficient is ±1.
**Graph Analysis:**
[Insert Graph Description]
The given graph shows a polynomial function intersecting the x-axis at multiple points. The graph appears to be a cubic polynomial with three real roots.
- The x-axis intersections (roots of the function) are approximately at x = -2, x = 1, and x = 3.
- The graph dips and rises, suggesting that it does not cross the x-axis in one smooth motion but has turning points indicating a polynomial with multiple terms.
**Graph Interpretation:**
The graph provided can be translated into a polynomial function in the following way:
1. Identify the roots of the function as seen on the x-axis crossings:
- Root 1: x = -2
- Root 2: x = 1
- Root 3: x = 3
2. Form the factors corresponding to each of these roots:
- For x = -2, the factor is (x + 2)
- For x = 1, the factor is (x - 1)
- For x = 3, the factor is (x - 3)
3. As the leading coefficient is either ±1, we can write the polynomial in factored form as:
\[
y = (x + 2)(x - 1)(x - 3)
\]
4. Verify the plotted graph's behavior, ensuring it matches the cubic nature with three intersections and corresponding changes in the slope.
**Answer Submission:**
`y = (x + 2)(x - 1)(x - 3)`
**Interactive Element:**
- [Submit Question Button]
- Allows students to verify their equation for correctness.
**Note:**
This exercise helps in understanding polynomial functions and how roots or zeros are identified and represented. The transformation of graphical behavior into a mathematical equation is a fundamental skill in algebra and calculus.
---
### Additional Resources:
- [Polynomial Functions: Concepts and Applications]
- [Graphing Techniques for Polynomial Equations]
- [Interactive Polynomial Graphing Tools]
### References:
- Polynomial graph interpretation and equation determination (Textbooks/Online Articles)
- Graphical tools used in educational settings (e.g., GeoGebra)
This example problem is designed for students
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