Characteristics of Polynomial Graphs

Front

What is a polynomial function?

Back

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where n is a non-negative integer.

Front

What determines the end behavior of a polynomial graph?

Back

The end behavior of a polynomial graph is determined by the degree of the polynomial and the sign of the leading coefficient.

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. It influences the direction of the graph as x approaches positive or negative infinity.

Front

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

Back

If the degree is even, both ends of the graph will point in the same direction. If the degree is odd, the ends will point in opposite directions.

Front

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

Back

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

Front

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

Back

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

Front

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

Back

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

Front

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

Back

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

Front

What is the significance of the degree of a polynomial?

Back

The degree of a polynomial indicates the maximum number of x-intercepts (real roots) and the general shape of the graph.

Front

How can you determine the end behavior of f(x) = 4x^3 - 2x + 1?

Back

Since the degree is odd and the leading coefficient is positive, as x → +∞, f(x) → +∞ and as x → -∞, f(x) → -∞.