Graphing Polynomials

The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity. It is determined by the leading term of the polynomial.

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It influences the end behavior of the graph.

A polynomial has an even degree if the highest exponent of its variable is an even number. The end behavior of even-degree polynomials is the same on both ends (either both go to +∞ or both go to -∞).

A polynomial has an odd degree if the highest exponent of its variable is an odd number. The end behavior of odd-degree polynomials is opposite on both ends (one goes to +∞ and the other to -∞).

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

Multiplicity refers to the number of times a particular root appears in a polynomial. A root with an even multiplicity will touch the x-axis, while a root with an odd multiplicity will cross the x-axis.

The factored form of a polynomial expresses it as a product of its linear factors. For example, $$f(x) = (x - r_1)(x - r_2)...(x - r_n)$$ where r are the roots.

The y-intercept is the point where the graph crosses the y-axis, found by evaluating f(0). It provides information about the polynomial's value at x = 0.

The degree of a polynomial can be identified by the number of turning points in the graph. A polynomial of degree n can have at most n-1 turning points.

A polynomial of degree n can have up to n real roots, counting multiplicities.