Polynomials - End Behavior and Graphs

Front

What is a polynomial?

Back

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, 3x^2 + 2x + 1 is a polynomial.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial 4x^3 + 2x^2 + 5, the degree is 3.

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. For example, in 5x^4 + 3x^2 + 2, the leading coefficient is 5.

Front

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

Back

The degree of a polynomial determines the end behavior of its graph. If the degree is even, both ends of the graph will either rise or fall together. If the degree is odd, one end will rise while the other falls.

Front

What is the end behavior of a polynomial with an even degree and a positive leading coefficient?

Back

The end behavior will be: as x → ∞, f(x) → ∞ and as x → -∞, f(x) → ∞.

Front

What is the end behavior of a polynomial with an odd degree and a positive leading coefficient?

Back

The end behavior will be: as x → ∞, f(x) → ∞ and as x → -∞, f(x) → -∞.

Front

What is the end behavior of a polynomial with an even degree and a negative leading coefficient?

Back

The end behavior will be: as x → ∞, f(x) → -∞ and as x → -∞, f(x) → -∞.

Front

What is the end behavior of a polynomial with an odd degree and a negative leading coefficient?

Back

The end behavior will be: as x → ∞, f(x) → -∞ and as x → -∞, f(x) → ∞.

Front

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

Back

To determine the end behavior, look at the degree and leading coefficient of the polynomial. This will indicate how the graph behaves as x approaches positive or negative infinity.

Front

What is the significance of the leading term in a polynomial?

Back

The leading term, which is the term with the highest degree, dominates the behavior of the polynomial for large values of x.