The power function

The two sides of $y=a{x}^2$ have the same height. Keep the exponent an integer from 2 upward, and let $a$ only turn the picture over. Raise that exponent through 3, 4, and 5. Which of those graphs still match on the left and the right, and which of them take opposite signs?

Both sides keep the same height

This picture keeps the exponent even, starting at 2, and starts with $a>0$.

The marks at inputs $1$ and $-1$ share the height $a$. While $a>0$ the arms open upward. That bowl, with its vertex at the origin, is the graph of a quadratic function $y=a{x}^2$. Drag the exponent through 4 and 6 and the marks stay level, so an even power keeps the same picture. Drag $a$ below zero and the bowl opens downward. At $a=0$ both marks lie on the axis.

The two sides take opposite signs

Freeze the exponent on an odd integer, starting at 3.

The mark at input $1$ has height $a$, and the mark at input $-1$ has height $-a$. While $a>0$ the graph rises through the origin, the same picture as $y=a{x}^3$. Drag the exponent through 5 and 7 and the heights stay opposite. Drag $a$ below zero and the signs swap.

The two pictures are one family. For an integer $n=2,3,\dots$,

$y=a{x}^n.$

When $n$ is even the graph matches $y=a{x}^2$. When $n$ is odd it matches $y=a{x}^3$. The sign of $a$ turns either picture over.

Even exponents keep one height on both sides. Odd exponents force opposite signs. The exponent starts at 2, so $n=1$ stays out of this family. The next question is what the graph does when the variable sits in the exponent, which is the Euler e-function.