To introduce to you, polynomials are algebraic equations containing more than two terms. The degree of a polynomial is determined by the term containing the highest exponent. When arranged from the highest to the lowest degree, the leading coefficient is the constant beside the term with the highest degree. An example would be: 2x² + 5x +6. The degree of this polynomial is 2 and the leading coefficient is also 2 from the term 2x².
For even-degree polynomials, the graphs starts from the left and ends to the right on the same direction. If the graph enters the graph from the up, the graph would also extend up to infinity. If the leading coefficient is positive, the graph starts and ends on the upward direction. When it's negative, it starts and ends below.
For odd-degree polynomials, the start and end of the graph are in opposite directions. If it starts from below, it will end extending upwards. When it comes to leading coefficients, a positive one would have a graph that starts downward and ending upwards. The opposite is true for the negative leading coefficients.
Please see the picture attached to better understand the descriptions.