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Consider the curve given by the equation

What symmetries does this curve display? Check all that
applicable.
symmetry about the -axis
symmetry about the -axis
no symmetry about the origin

Ask by Davison Riley. in the United States
Mar 28,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

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The curve is symmetric about the origin.

Solución

We start with the equation
  1. To check for symmetry about the -axis, replace with :
This is not equivalent to the original equation since the sign of the term has changed.
  1. To check for symmetry about the -axis, replace with :
Again, this does not match the original equation because the term has a different sign.
  1. To check for symmetry about the origin, replace with :
This is exactly the original left‐hand side of the equation, so the curve is symmetric about the origin.
Thus, the curve is not symmetric about the -axis or the -axis, and it is symmetric about the origin. Since the provided options are:
  • symmetry about the -axis,
  • symmetry about the -axis,
  • no symmetry about the origin,
none of these choices apply because the curve actually has symmetry about the origin.

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The Deep Dive

To analyze the symmetries of the curve defined by the equation , we can apply some tests.
First, to check for symmetry about the -axis, we can substitute with in the equation. If the resulting equation is equivalent to the original, then the curve has symmetry about the -axis. Upon substituting, we find that this leads to a different expression, confirming no symmetry about the -axis.
Next, testing for symmetry about the -axis involves substituting with . If the equation remains unchanged, then the curve has symmetry about the -axis. In this case, substituting reveals that the equation remains unchanged, indicating symmetry about the -axis.
Lastly, to check for symmetry about the origin, we can substitute both and with their negatives. If the resulting equation equals the original, this shows symmetry about the origin. However, this substitution results in a different expression, confirming no symmetry about the origin.
From these findings, we conclude that the curve exhibits symmetry about the -axis, but not about the -axis or the origin.
Thus, the applicable symmetries are:
  • symmetry about the -axis
  • no symmetry about the origin

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