2024 AMC 12A Problem 13

Attempt Problem 13 of the 2024 AMC 12A below, then check your answer against the professionally curated solution 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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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)

The axis of symmetry is the vertical line

x=c

where the function attains its minimum

Set the derivative

e^{x+1}-e^{-x}=0

to find

c,

then reflect

x=-1

across it

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,

x=-\tfrac12.

Reflecting

\left(-1,\tfrac12\right)

across

x=-\tfrac12

keeps the

y

-coordinate and sends

x=-1

x=0.

The image is

\left(0,\tfrac12\right).

Thus, the correct answer is D.

Problem 13 in other years