3.3.4 End Behavior and Multiplicity

Front

What is the end behavior of a polynomial function?

Back

The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity, determined by the degree and leading coefficient.

Front

How does the degree of a polynomial affect its end behavior?

Back

The degree of a polynomial determines the number of times the graph can turn, influencing whether the ends of the graph go up or down.

Front

What is the leading coefficient of a polynomial?

Back

The leading coefficient is the coefficient of the term with the highest degree in a polynomial, which influences the direction of the graph's end behavior.

Front

If a polynomial has an odd degree and a negative leading coefficient, what is its end behavior?

Back

As x → ∞, f(x) → -∞ and as x → -∞, f(x) → ∞.

Front

If a polynomial has an even degree and a positive leading coefficient, what is its end behavior?

Back

As x → ∞, f(x) → ∞ and as x → -∞, f(x) → ∞.

Front

What does it mean for a polynomial to have a degree of 4?

Back

A polynomial of degree 4 can have up to 4 real roots and its graph can have up to 3 turning points.

Front

What is the significance of the leading coefficient being positive?

Back

A positive leading coefficient indicates that the ends of the graph will rise as x approaches both positive and negative infinity for even-degree polynomials.

Front

What is the significance of the leading coefficient being negative?

Back

A negative leading coefficient indicates that the ends of the graph will fall as x approaches both positive and negative infinity for even-degree polynomials.

Front

Describe the end behavior of the polynomial f(x) = -3x^5 + 2x^3 - x + 1.

Back

As x → ∞, f(x) → -∞ and as x → -∞, f(x) → ∞.

Front

What is the end behavior of the polynomial f(x) = 2x^6 - 4x^2 + 1?

Back

As x → ∞, f(x) → ∞ and as x → -∞, f(x) → ∞.