End Behavior & Zeros of Polynomials

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.

Front

How do you determine the end behavior of a polynomial function?

Back

To determine the end behavior, look at the leading term of the polynomial. The sign and degree of the leading term dictate the behavior as x approaches ±∞.

Front

What does it mean if a polynomial has an odd degree and a negative leading coefficient?

Back

If a polynomial has an odd degree and a negative leading coefficient, as x → ∞, f(x) → -∞ and as x → -∞, f(x) → ∞.

Front

What does it mean if a polynomial has an even degree and a positive leading coefficient?

Back

If a polynomial has an even degree and a positive leading coefficient, as x → ∞, f(x) → ∞ and as x → -∞, f(x) → ∞.

Front

What are the zeros of a polynomial function?

Back

The zeros of a polynomial function are the values of x for which the function equals zero. They are the x-intercepts of the graph.

Front

How do you find the zeros of the polynomial y = x(x - 6)(x + 5)?

Back

Set the polynomial equal to zero: x(x - 6)(x + 5) = 0. The zeros are x = 0, x = 6, and x = -5.

Front

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

Back

The leading coefficient determines the direction of the graph as x approaches ±∞. A positive leading coefficient means the graph rises to the right, while a negative one means it falls.

Front

What is the end behavior of f(x) = -7x^3 + 2x - 1?

Back

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

Front

What is the end behavior of f(x) = 3x^4 + 2x - 5?

Back

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

Front

What is the relationship between the degree of a polynomial and its end behavior?

Back

The degree of a polynomial indicates the number of times the graph can change direction. Odd degrees can have different end behaviors on either side, while even degrees have the same end behavior.