Question
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.
A. As
B. As
C. As
D. As
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.
-
As
approaches positive infinity ( ): - The term
will approach 0 because . Therefore:
- The term
-
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:
- The term
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.
Answered by UpStudy AI and reviewed by a Professional Tutor
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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.