Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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### 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
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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
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