Homework Help is Here – Start Your Trial Now!
Math
Calculus
Which of the following matches the graph of f(x) = 2(x − 1)(x + + 1)²(x + 2)?

Which of the following matches the graph of f(x) = 2(x − 1)(x + + 1)²(x + 2)?

Calculus: Early Transcendentals
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...
See similar textbooks
icon
Related questions
Question
### Analyzing Polynomial Graphs **Question:** Which of the following matches the graph of \( f(x) = 2(x - 1)(x + 1)^2(x + 2)? \) **Explanation:** We start by analyzing the given polynomial function \( f(x) = 2(x - 1)(x + 1)^2(x + 2) \). The degrees of the polynomial and the roots can be identified to understand the possible shape of the graph. - The roots are \(x = 1\), \(x = -1\) (which is a double root), and \(x = -2\). - At \(x = 1\): The graph will cross the x-axis. - At \(x = -1\): The graph will touch the x-axis and turn around since \(x = -1\) is a double root. - At \(x = -2\): The graph will cross the x-axis. ### Graph 1: **Description:** The graph shows a polynomial curve with the following characteristics: - Crosses the x-axis at \(x = 1\) and \(x = -2\). - Touches the x-axis at \(x = -1\) and turns back up, indicating a double root. - The graph follows a consistent end behavior as \( x \to -\infty \) and \(x \to +\infty\). ### Graph 2: **Description:** This graph showcases a different polynomial curve: - Crosses the x-axis at multiple points. - The behavior at the roots isn't consistent with a double root at \(x = -1\), as it doesn't solely touch and turn at this point. ### Detailed Analysis: The correct graph should: - Cross the x-axis at \(x = 1\) and \(x = -2\). - Touch and turn at \(x = -1\) indicating a double root at this point. ### Conclusion: **Graph 1** accurately represents the graph of \(f(x) = 2(x - 1)(x + 1)^2(x + 2)\). It correctly displays the polynomial touching and turning at \(x = -1\) (double root), and crossing at \(x = 1\) and \(x = -2\). ### Additional Notes: - Understanding the nature of roots and their multiplicities is crucial for sketching polynomial
Transcribed Image Text:### Analyzing Polynomial Graphs **Question:** Which of the following matches the graph of \( f(x) = 2(x - 1)(x + 1)^2(x + 2)? \) **Explanation:** We start by analyzing the given polynomial function \( f(x) = 2(x - 1)(x + 1)^2(x + 2) \). The degrees of the polynomial and the roots can be identified to understand the possible shape of the graph. - The roots are \(x = 1\), \(x = -1\) (which is a double root), and \(x = -2\). - At \(x = 1\): The graph will cross the x-axis. - At \(x = -1\): The graph will touch the x-axis and turn around since \(x = -1\) is a double root. - At \(x = -2\): The graph will cross the x-axis. ### Graph 1: **Description:** The graph shows a polynomial curve with the following characteristics: - Crosses the x-axis at \(x = 1\) and \(x = -2\). - Touches the x-axis at \(x = -1\) and turns back up, indicating a double root. - The graph follows a consistent end behavior as \( x \to -\infty \) and \(x \to +\infty\). ### Graph 2: **Description:** This graph showcases a different polynomial curve: - Crosses the x-axis at multiple points. - The behavior at the roots isn't consistent with a double root at \(x = -1\), as it doesn't solely touch and turn at this point. ### Detailed Analysis: The correct graph should: - Cross the x-axis at \(x = 1\) and \(x = -2\). - Touch and turn at \(x = -1\) indicating a double root at this point. ### Conclusion: **Graph 1** accurately represents the graph of \(f(x) = 2(x - 1)(x + 1)^2(x + 2)\). It correctly displays the polynomial touching and turning at \(x = -1\) (double root), and crossing at \(x = 1\) and \(x = -2\). ### Additional Notes: - Understanding the nature of roots and their multiplicities is crucial for sketching polynomial
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 8 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS