Difference between revisions of "2024 AMC 12A Problems/Problem 13"
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− | Consider the graphs of <math>y=e^{x+1}-1</math> and <math>y=e^{-x}-1. A rough sketch will show that they intercept somewhere between -1 and 0 and the axis of symmetry is vertical. Thus, < | + | Consider the graphs of <math>y=e^{x+1}-1</math> and <math>y=e^{-x}-1</math>. A rough sketch will show that they intercept somewhere between -1 and 0 and the axis of symmetry is vertical. Thus, <math>\boxed{\textbf{(D) }\left(0,\dfrac12\right)}.</math> is the only possible answer. |
− | Note: You can more rigorously think about the solution by noting that since that the absolute value of the derivative of the power that e is raised to is the same, and they are both subtracted by 1, then for all x greater than the point of interception, both equations will grow by the same amount. Setting both equations equal to each other, it is trivial to see | + | Note: You can more rigorously think about the solution by noting that since that the absolute value of the derivative of the power that e is raised to is the same, and they are both subtracted by 1, then for all x greater than the point of interception, both equations will grow by the same amount. Setting both equations equal to each other, it is trivial to see $x=-1/2, giving us the axis of symmetry. |
~woeIsMe | ~woeIsMe |
Revision as of 19:35, 8 November 2024
Contents
[hide]Problem
The graph of has an axis of symmetry. What is the reflection of the point over this axis?
Solution 1
The line of symmetry is probably of the form for some constant . A vertical line of symmetry at for a function exists if and only if ; we substitute and into our given function and see that we must have
for all real . Simplifying:
If , then for all real ; this is clearly impossible, so let . Thus, our line of symmetry is , and reflecting over this line gives
~Technodoggo
Solution 2
Consider the graphs of and . A rough sketch will show that they intercept somewhere between -1 and 0 and the axis of symmetry is vertical. Thus, is the only possible answer.
Note: You can more rigorously think about the solution by noting that since that the absolute value of the derivative of the power that e is raised to is the same, and they are both subtracted by 1, then for all x greater than the point of interception, both equations will grow by the same amount. Setting both equations equal to each other, it is trivial to see $x=-1/2, giving us the axis of symmetry.
~woeIsMe
See also
2024 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.