Understanding Degree 4 and 5 Polynomial Functions

TLDR; Polynomial functions have x-intercepts, turning points, and end behavior related to degree and leading coefficient.

📈 Polynomial Function Information

Polynomial functions provide information about the graph of the function.

A polynomial function of degree 'n' has at most 'n' x-intercepts and at most 'n-1' turns.

The number of x-intercepts and turns is directly related to the degree of the polynomial function.

📊 Degree 4 Polynomial Function

A degree 4 polynomial function can have at most 4 x-intercepts and at most 3 turns.

The function's graph reflects the number of x-intercepts and turns according to the degree.

The turning points of the graph correspond to either high points or low points, giving maximum or minimum function values.

🔄 End Behavior of Degree 4 Polynomial

For a degree 4 polynomial function with a positive leading coefficient, the graph moves up at both positive and negative infinity, resulting in f(x) approaching positive infinity.

When the leading coefficient is negative, the graph moves down at both infinities, causing f(x) to approach negative infinity in both cases.

📈 Degree 5 Polynomial Function

A degree 5 polynomial function can have at most 5 x-intercepts and at most 4 turns.

The number of x-intercepts and turns is directly determined by the degree of the polynomial function.

🔍 Degree 5 Polynomial Function Variations

A degree 5 polynomial function may have fewer x-intercepts and turns, making it appear to be of a lower degree.

The equation of a polynomial function is essential in determining its actual degree.

📉 End Behavior of Odd Degree Polynomial

Odd degree polynomial functions with a negative leading coefficient result in f(x) approaching negative infinity as x moves to the right and approaching positive infinity as x moves to the left.

The opposite end behavior occurs when the leading coefficient is positive.

🔚 Review of End Behavior

The end behavior of polynomial functions with odd degree and different leading coefficients was reviewed as a conclusion.

This serves as a review of the long run behavior of polynomial functions.