2024 AMC 12A Problem 13

Below is the professionally curated solution for Problem 13 of the 2024 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 12A solutions, or check the answer key.

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Concepts:function symmetry

Difficulty rating: 1660

13.

The graph of $y=e^{x+1}+e^{-x}-2$ has an axis of symmetry. What is the reflection of the point $\left(-1,\tfrac12\right)$ over this axis?

$\left(-1,-\tfrac32\right)$

$(-1,0)$

$\left(-1,\tfrac12\right)$

$\left(0,\tfrac12\right)$

$\left(3,\tfrac12\right)$

Solution:

The curve $y=e^{x+1}+e^{-x}-2$ is symmetric about the vertical line through its minimum. Setting the derivative $e^{x+1}-e^{-x}=0$ gives $x+1=-x,$ so $x=-\tfrac12.$ Reflecting $\left(-1,\tfrac12\right)$ across $x=-\tfrac12$ keeps the $y$ -coordinate and sends $x=-1$ to $x=0.$ The image is $\left(0,\tfrac12\right).$ Thus, the correct answer is D.

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