When we look at polynomial functions, two key things affect how they act: the degree of the polynomial and the leading coefficient. Knowing about these can help us guess how a polynomial will look when we draw it.

Degree of a Polynomial:

1. What It Is: The degree of a polynomial is the highest exponent (or power) of the variable in the equation.

2. How It Affects the Graph:

   • For even degrees: Both ends of the graph will either go up or down together.
     • If the leading coefficient is positive, the graph will go up on both sides.
     • If it’s negative, it will go down on both sides.
   • For odd degrees: The ends will go in opposite directions.
     • If the leading coefficient is positive, the left end will go down while the right end goes up.
     • If it’s negative, the left end will go up and the right end will go down.

Leading Coefficient:

1. What It Is: The leading coefficient is the number in front of the term with the highest degree.

2. How It Changes the Graph:

   • A positive leading coefficient means the graph will open upwards (for even degrees) or rise on the right side (for odd degrees).
   • A negative leading coefficient means the graph opens downwards (for even degrees) or falls on the right side (for odd degrees).

To Sum It Up:

• The degree of the polynomial helps us understand the basic shape and how the ends of the graph will behave.

• The leading coefficient helps us see if the graph opens up or down and how steeply it rises or falls.

Knowing these two things can make working with polynomial functions much easier and clearer!