Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Determine the End Behavior of the Polynomial Function**
Consider the polynomial function given by:
\[ h(w) = -4w^3 - 3w^4 - w^6 - 5w^5 \]
We need to determine the end behavior of this function.
**Options for End Behavior:**
a. Two arrows pointing upwards \(\uparrow \uparrow\).\
b. Two arrows pointing downwards \(\downarrow \downarrow\).\
c. Left arrow pointing downwards and right arrow pointing upwards \(\downarrow \uparrow\).\
d. Left arrow pointing upwards and right arrow pointing downwards \(\uparrow \downarrow\).
**Explanation:**
Examining each option:
- **Option a**: \( \uparrow \uparrow \)
- **Option b**: \( \downarrow \downarrow \)
- **Option c**: \( \downarrow \uparrow \)
- **Option d**: \( \uparrow \downarrow \)
When examining the highest degree term of the polynomial, which is \( -w^6 \), we can determine the end behavior:
The term \( -w^6 \) shows that it is a 6th degree polynomial with a negative leading coefficient. For this type of polynomial:
- As \( w \to -\infty \), \( h(w) \to -\infty \)
- As \( w \to \infty \), \( h(w) \to -\infty \)
**Conclusion:**
Therefore, the correct end behavior of \[ h(w) = -4w^3 - 3w^4 - w^6 - 5w^5 \] is represented by option b., where both arrows point downwards \(\downarrow \downarrow\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9810359e-e211-4db8-9482-921b0ee941bb%2F53ed5ada-f919-4996-be26-a88dbd642924%2Fkojvndf_processed.png&w=3840&q=75)
Transcribed Image Text:**Determine the End Behavior of the Polynomial Function**
Consider the polynomial function given by:
\[ h(w) = -4w^3 - 3w^4 - w^6 - 5w^5 \]
We need to determine the end behavior of this function.
**Options for End Behavior:**
a. Two arrows pointing upwards \(\uparrow \uparrow\).\
b. Two arrows pointing downwards \(\downarrow \downarrow\).\
c. Left arrow pointing downwards and right arrow pointing upwards \(\downarrow \uparrow\).\
d. Left arrow pointing upwards and right arrow pointing downwards \(\uparrow \downarrow\).
**Explanation:**
Examining each option:
- **Option a**: \( \uparrow \uparrow \)
- **Option b**: \( \downarrow \downarrow \)
- **Option c**: \( \downarrow \uparrow \)
- **Option d**: \( \uparrow \downarrow \)
When examining the highest degree term of the polynomial, which is \( -w^6 \), we can determine the end behavior:
The term \( -w^6 \) shows that it is a 6th degree polynomial with a negative leading coefficient. For this type of polynomial:
- As \( w \to -\infty \), \( h(w) \to -\infty \)
- As \( w \to \infty \), \( h(w) \to -\infty \)
**Conclusion:**
Therefore, the correct end behavior of \[ h(w) = -4w^3 - 3w^4 - w^6 - 5w^5 \] is represented by option b., where both arrows point downwards \(\downarrow \downarrow\).
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