Understanding how the degree and leading coefficient affect polynomial graphs is key to knowing their features. Let's break down these important parts.

1. Degree of the Polynomial

The degree of a polynomial is the highest power of the variable used. For example, in the polynomial $P(x) = 3x^4 + 2x^3 - x + 7$, the degree is 4. The degree greatly affects the shape and how the graph behaves at the ends:

• Odd Degree Polynomials (like degree 1, 3, or 5):

  • End behavior: If the leading coefficient is positive, as $x$ gets really big (x goes to infinity), $y$ also gets really big. If it's negative, $y$ will get really small (y goes to negative infinity).
  • Example: The graph of $P(x) = x^3$ rises to the right and falls to the left.

• Even Degree Polynomials (like degree 0, 2, or 4):

  • End behavior: Both ends of the graph go up or down together, depending on the leading coefficient.
  • Example: For $Q(x) = x^4$, both sides of the graph rise, making a "U" shape.

2. Leading Coefficient

The leading coefficient is the number in front of the highest degree term in a polynomial. It changes how the graph looks in terms of stretch and direction.

• Positive Leading Coefficient:

  • The graph opens upwards.
  • Example: In $P(x) = 2x^4$, the graph opens upward, looking like a "cup."

• Negative Leading Coefficient:

  • The graph opens downwards.
  • Example: In $Q(x) = -x^3$, the graph opens downward, looking like a "frown."

Putting It All Together

To guess the overall shape of a polynomial graph, look at both the degree and the leading coefficient. For example, a polynomial with a degree of 4 and a positive leading coefficient will look like a "smiling" U, while a polynomial with a degree of 3 and a negative leading coefficient will look like a "frowning" curve.