2024 AMC 12A Problems/Problem 13

Problem

The graph of $y=e^{x+1}+e^{-x}-2$ has an axis of symmetry. What is the reflection of the point $(-1,\tfrac{1}{2})$ over this axis?

$\textbf{(A) }\left(-1,-\frac{3}{2}\right)\qquad\textbf{(B) }(-1,0)\qquad\textbf{(C) }\left(-1,\tfrac{1}{2}\right)\qquad\textbf{(D) }\left(0,\frac{1}{2}\right)\qquad\textbf{(E) }\left(3,\frac{1}{2}\right)$

Solution 1

The line of symmetry is probably of the form $x=a$ for some constant $a$ . A vertical line of symmetry at $x=a$ for a function $f$ exists if and only if $f(a-b)=f(a+b)$ ; we substitute $a-b$ and $a+b$ into our given function and see that we must have

$e^{a-b+1}+e^{-(a-b)}-2=e^{a+b+1}+e^{-(a+b)}-2$

for all real $b$ . 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 $e^{a+1}-e^{-a}\neq0$ , then $e^{-b}=e^b$ for all real $b$ ; this is clearly impossible, so let $e^{a+1}-e^{-a}=0\implies a+1=-a\implies a=-\dfrac12$ . Thus, our line of symmetry is $x=-\dfrac12$ , and reflecting $\left(-1,\dfrac12\right)$ over this line gives $\boxed{\textbf{(D) }\left(0,\dfrac12\right)}.$

~Technodoggo

Solution 2

Consider the graphs of $y=e^{x+1}-1$ and $y=e^{-x}-1. A rough sketch will show that they intercept somewhere between -1 and 0 and the axis of symmetry is vertical. Thus,$ \boxed{\textbf{(D) }\left(0,\dfrac12\right)}.$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$ (Error compiling LaTeX. Unknown error_msg)x=-1/2, giving us the axis of symmetry.

~woeIsMe

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
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2024 AMC 12A Problems/Problem 13

Contents

Problem

Solution 1

Solution 2

See also