End behavior and asymptotes are important when figuring out what polynomial graphs look like.

End Behavior:

• The leading term of a polynomial, which is the highest power of $x$, helps us know how the graph behaves as $x$ goes really high (toward infinity) or really low (toward negative infinity).

  • For instance, with a polynomial like $f(x) = 2x^3$, the graph will go up high on both sides. But for $f(x) = -3x^2$, the graph will go down as $x$ increases to infinity or decreases to negative infinity.

Asymptotes:

• Polynomials don't have vertical or horizontal asymptotes like some other types of functions do. But looking at their limits and end behavior can help us see where the graph flattens out or turns.

By exploring these ideas, students can draw better and more detailed polynomial graphs!