Question
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Use the leading coefficient test to determine the end behavior of the graph of the given polynomial function.

A. Falls left rises right.
B. Falls left falls right.
C. Rises left rises right.
D. Rises left falls right.
E. None of the above.

Ask by Campos Huang. in the United States
Nov 02,2024

Upstudy AI Solution

Tutor-Verified Answer

Answer

The end behavior is falls left and rises right.

Solution

To determine the end behavior of the polynomial function using the leading coefficient test, we need to look at the degree of the polynomial and the sign of the leading coefficient.
  1. Degree of the polynomial: The highest power of in the polynomial is , which means the degree is .
  2. Leading coefficient: The leading coefficient is the coefficient of the term with the highest degree, which in this case is .
Now, we can analyze the end behavior based on the degree and the leading coefficient:
  • Since the degree is odd, the end behavior will be:
    • As (left side), will go to .
    • As (right side), will go to .
  • The leading coefficient is positive, which confirms that the function rises to the right.
Putting this together, we find that the end behavior of the graph of the polynomial function is:
  • Falls left (as )
  • Rises right (as )
Thus, the correct answer is:
A. Falls left & rises right.

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To determine the end behavior of the polynomial function , we focus on the leading term, which is . Since the degree (7) is odd and the leading coefficient (7) is positive, the end behavior of the graph will be that it falls to the left and rises to the right. Therefore, the correct answer is A. Falls left & rises right.
Additionally, the degree of a polynomial not only influences its end behavior but also reveals information about the number of turning points. An odd-degree polynomial can have a maximum of turning points. In this case, since it’s a 7th-degree polynomial, it can have up to six turning points. So expect some twists and turns along the graph!

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