2024 AMC 12A Problems/Problem 13

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:

\begin{align*} e^{a-b+1}+e^{-(a-b)}-2&=e^{a+b+1}+e^{-(a+b)}-2 \\ e^{a-b+1}+e^{b-a}&=e^{a+b+1}+e^{-a-b} \\ e^{a-b+1}-e^{-a-b}&=e^{a+b+1}-e^{b-a} \\ e^{-b}\left(e^{a+1}-e^{-a}\right)&=e^b\left(e^{a+1}-e^{-a}\right). \\ \end{align*}

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 cheese)

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 the derivative of the power that e is raised to in one equation is equal to the derivative of the power that e is raised to multiplied by , and both equations are subtracted by 1, then the sum of both equations 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

=Solution 3(Comparing, subsiution and Derivatives)

See also

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