When we look at polynomial functions, the differences between even and odd degree polynomials can be really interesting! Here are some patterns I've noticed:

Even Degree Polynomials:

• End Behavior: Both ends go in the same direction.

  • If the leading number is positive, the ends rise up to infinity on both sides.
  • If it's negative, the ends fall down to infinity.

• Symmetry: They often look the same on both sides of the y-axis, which is why we call them "even."

• Number of Roots: They can have up to $n$ real roots, where $n$ is the degree of the polynomial. Sometimes, there might be no real roots at all.

Odd Degree Polynomials:

• End Behavior: The ends go in opposite directions.

  • If the leading number is positive, it goes up on the right side and down on the left side.

• Asymmetry: They usually do not look the same on both sides, which is why we call them "odd."

• Number of Roots: They always have at least one real root because of something called the Intermediate Value Theorem.

These features make it fun to study polynomial graphs!