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Describe the end behavior of the graph of: ƒ (x) = −5x5 + 4x² + 12x² - 8 using limits. —

Describe the end behavior of the graph of: ƒ (x) = −5x5 + 4x² + 12x² - 8 using limits. —

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
**Title: Understanding the End Behavior of Polynomial Functions** **Introduction:** When analyzing polynomial functions, it's essential to understand their end behavior. This behavior can be described using limits, giving insight into how the function behaves as the input values become very large or very small. Let's examine the end behavior of the following polynomial function. **Function:** \( f(x) = -5x^5 + 4x^4 + 12x^2 - 8 \) **Objective:** Describe the end behavior of the graph of \( f(x) \) using limits. **Instructions:** - **As \( x \to \infty \):** Determine what happens to \( f(x) \). - **As \( x \to -\infty \):** Determine what happens to \( f(x) \). **Explanation:** For a polynomial function, the highest degree term determines the end behavior. Here, \( -5x^5 \) is the leading term. Since it has an odd degree (5) and a negative coefficient, the end behavior is as follows: - As \( x \to \infty \), \( f(x) \to -\infty \). - As \( x \to -\infty \), \( f(x) \to \infty \). By analyzing these trends, one can sketch the overall shape of the graph and predict its behavior at the extremes.
Transcribed Image Text:**Title: Understanding the End Behavior of Polynomial Functions** **Introduction:** When analyzing polynomial functions, it's essential to understand their end behavior. This behavior can be described using limits, giving insight into how the function behaves as the input values become very large or very small. Let's examine the end behavior of the following polynomial function. **Function:** \( f(x) = -5x^5 + 4x^4 + 12x^2 - 8 \) **Objective:** Describe the end behavior of the graph of \( f(x) \) using limits. **Instructions:** - **As \( x \to \infty \):** Determine what happens to \( f(x) \). - **As \( x \to -\infty \):** Determine what happens to \( f(x) \). **Explanation:** For a polynomial function, the highest degree term determines the end behavior. Here, \( -5x^5 \) is the leading term. Since it has an odd degree (5) and a negative coefficient, the end behavior is as follows: - As \( x \to \infty \), \( f(x) \to -\infty \). - As \( x \to -\infty \), \( f(x) \to \infty \). By analyzing these trends, one can sketch the overall shape of the graph and predict its behavior at the extremes.
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