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, 2x^2 + 3x + 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 - x + 7, 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 leading coefficient affect the end behavior of a polynomial?

Back

If the leading coefficient is positive and the degree is even, the ends of the graph rise in both directions. If the leading coefficient is positive and the degree is odd, the left end falls and the right end rises.

Front

What is the end behavior of a polynomial function?

Back

End behavior describes how the values of a polynomial function behave as x approaches positive or negative infinity. It is determined by the degree and leading coefficient.

Front

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

Back

As x approaches -∞, f(x) approaches +∞; as x approaches +∞, f(x) approaches -∞.

Front

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

Back

As x approaches -∞, f(x) approaches +∞; as x approaches +∞, f(x) approaches +∞.

Front

How do you determine the degree of a polynomial?

Back

To determine the degree of a polynomial, identify the term with the highest exponent of the variable. The degree is the value of that exponent.

Front

What is the difference between even and odd degree polynomials in terms of end behavior?

Back

Even degree polynomials have the same end behavior on both sides (both ends rise or both ends fall), while odd degree polynomials have opposite end behavior (one end rises and the other falls).

Front

What is a zero of a polynomial?

Back

A zero of a polynomial is a value of x that makes the polynomial equal to zero. For example, if f(x) = x^2 - 4, then x = 2 and x = -2 are zeros.

Tags

CCSS.HSF-IF.C.7C