End Behavior

Front

What is end behavior in the context of polynomial functions?

Back

End behavior describes how the values of a function behave as the input (x) approaches positive or negative infinity.

Front

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

Back

The degree of a polynomial determines the direction of the graph as x approaches infinity or negative infinity. Even degrees lead to the same behavior at both ends, while odd degrees lead to opposite behaviors.

Front

What does a positive leading coefficient indicate about the end behavior of a polynomial?

Back

If the leading coefficient is positive and the degree is even, the graph rises to infinity on both ends. If the degree is odd, the graph falls to negative infinity on the left and rises to positive infinity on the right.

Front

What does a negative leading coefficient indicate about the end behavior of a polynomial?

Back

If the leading coefficient is negative and the degree is even, the graph falls to negative infinity on both ends. If the degree is odd, the graph rises to positive infinity on the left and falls to negative infinity on the right.

Front

What is the end behavior of the function f(x) = -2x^7 + 2x^4 + 4?

Back

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

Front

What is the degree of the polynomial function f(x) = 4x^5 - 4x^4 + 2x^3 - 9?

Back

The degree of the polynomial is 5.

Front

What is the leading coefficient of the polynomial function f(x) = 3x^8 + 3x^5 - 4x + 7?

Back

The leading coefficient is 3, which is positive.

Front

Describe the end behavior of the graph of a polynomial with an odd degree and a negative leading coefficient.

Back

As x → ∞, y → -∞ and as x → -∞, y → ∞.

Front

Describe the end behavior of the graph of a polynomial with an even degree and a positive leading coefficient.

Back

As x → ∞, y → ∞ and as x → -∞, y → ∞.

Front

What is the significance of the leading term in determining end behavior?

Back

The leading term, which is the term with the highest degree, dominates the behavior of the polynomial as x approaches infinity or negative infinity.