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</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. | 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 | + | 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 the sum of both functions will be the same from one side of the interception to the other. Setting both equations equal to each other, it is trivial to see <math>x=-1/2</math>, giving us the axis of symmetry. |
(someone insert the graph pls) | (someone insert the graph pls) |
Revision as of 19:43, 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 (Graphing)
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 the sum of both functions will be the same from one side of the interception to the other. Setting both equations equal to each other, it is trivial to see , giving us the axis of symmetry.
(someone insert the graph pls)
~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.