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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![On the educational website, include the following text and explanation of the graph:
---
### Identifying the Degree of a Polynomial from its Graph
**Graph Analysis:**
Below is a graph of a polynomial function:
[Graph Image]
The graph shows the curve of a polynomial that intersects the x-axis at several points and exhibits multiple turning points or changes in direction.
1. **Turning Points:** The graph has 3 turning points visible. Turning points are where the graph changes direction from increasing to decreasing or vice versa.
2. **Intersections with the X-axis:** The graph crosses the x-axis at three distinct points (these are the roots of the polynomial).
**Question:**
What is the least possible degree of the polynomial graphed above?
**Answer Input Box:**
---
**Explanation:**
The least possible degree of a polynomial can be determined by the number of turning points in its graph. The number of turning points is at most \(n-1\) for a polynomial of degree \(n\).
Since there are 3 turning points in the graph, the least possible degree \(n\) of the polynomial is \(n-1 \geq 3\), which simplifies to \(n \geq 4\).
Thus, the polynomial is at least of degree 4.
---
Make sure to provide this background explanation so students can understand the reasoning behind determining the degree of the polynomial from the graph.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6010960e-f970-4bb2-b167-12f0ced798a6%2Fa30b746a-f260-435a-9d7d-3cab8559aeaf%2Fifuih1.png&w=3840&q=75)
Transcribed Image Text:On the educational website, include the following text and explanation of the graph:
---
### Identifying the Degree of a Polynomial from its Graph
**Graph Analysis:**
Below is a graph of a polynomial function:
[Graph Image]
The graph shows the curve of a polynomial that intersects the x-axis at several points and exhibits multiple turning points or changes in direction.
1. **Turning Points:** The graph has 3 turning points visible. Turning points are where the graph changes direction from increasing to decreasing or vice versa.
2. **Intersections with the X-axis:** The graph crosses the x-axis at three distinct points (these are the roots of the polynomial).
**Question:**
What is the least possible degree of the polynomial graphed above?
**Answer Input Box:**
---
**Explanation:**
The least possible degree of a polynomial can be determined by the number of turning points in its graph. The number of turning points is at most \(n-1\) for a polynomial of degree \(n\).
Since there are 3 turning points in the graph, the least possible degree \(n\) of the polynomial is \(n-1 \geq 3\), which simplifies to \(n \geq 4\).
Thus, the polynomial is at least of degree 4.
---
Make sure to provide this background explanation so students can understand the reasoning behind determining the degree of the polynomial from the graph.
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