Question
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Describe the end behavior of the given function.

A. As decreases, approaches 2 .
B. As decreases, approaches negative infinity.
C. As increases, approaches -2 .
D. As increases, approaches infinity.

Ask by Sullivan Rodriquez. in the United States
Jan 31,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

As increases, approaches -2, and as decreases, approaches positive infinity. So, the correct options are C and B.

Solution

To analyze the end behavior of the function , we need to consider the limits of as approaches positive and negative infinity.
  1. As approaches positive infinity ( ):
    • The term will approach 0 because . Therefore:
  2. As approaches negative infinity ( ):
    • The term will approach infinity because raising a fraction less than 1 to a negative exponent results in a large positive number. Therefore:
Now, summarizing the end behavior:
  • As , .
  • As , .
Based on this analysis, the correct options are:
  • C. As increases, approaches -2.
  • B. As decreases, approaches positive infinity.
Thus, the correct answers are C and B.

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The Deep Dive

As approaches negative infinity, grows very large because raising a fraction to a negative exponent flips it into a positive exponent. In this case, as decreases, will approach positive infinity, meaning the correct choice is aligned with infinity’s pull rather than any finite number.
On the flip side, as approaches positive infinity, will shrink towards zero. Therefore, will approach -2, because the function’s structure reveals that all the action is pulling down towards a threshold just below zero, making it clear that our function heads towards this value as increases.

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