Chapter 1.2, Problem 90AYU

To determine

To explain: Why, if two of these symmetries are present, the remaining one must also be present.

Answer to Problem 90AYU

For $x\text{-axis}$ reflection symmetry, $f\left(x,y\right) = f\left(x,-y\right)$

For $y\text{-axis}$ reflection symmetry, $f\left(x,y\right) = f\left(-x,y\right)$

For symmetry about the origin, $f\left(x,y\right) = f\left(-x,-y\right)$

If you reflect about both $x$ and $y\text{-axis}$, it is equivalent to reflecting about the origin.

If you reflect about the origin and then one axis, you have reflected about the other axis.

Explanation of Solution

Given: An equation is being tested for symmetry with respect to the x\text{-axis}, the $y\text{-axis}$, and the origin.

Calculation:

For $x\text{-axis}$ reflection symmetry, $f\left(x,y\right) = f\left(x,-y\right)$

For $y\text{-axis}$ reflection symmetry, $f\left(x,y\right) = f\left(-x,y\right)$

For symmetry about the origin, $f\left(x,y\right) = f\left(-x,-y\right)$

If you reflect about both $x$ and $y\text{-axis}$, it is equivalent to reflecting about the origin.

If you reflect about the origin and then one axis, you have reflected about the other axis.