Graphing Polynomials

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to whole number powers and coefficients. For example, $$2x^3 - 4x^2 + 3x - 5$$ is a polynomial.

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $$3x^4 + 2x^3 - x + 7$$ , the degree is 4.

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. For example, in $$5x^3 + 2x^2 - 3$$ , the leading coefficient is 5.

A polynomial is increasing on an interval if its derivative is positive on that interval, and decreasing if its derivative is negative.

The sign of the leading coefficient determines the end behavior of the polynomial. If it is positive and the degree is even, the graph rises on both ends; if it is negative, it falls on both ends.

An even function is a function that satisfies the condition $$f(-x) = f(x)$$ for all x in its domain. Its graph is symmetric about the y-axis.

An odd function is a function that satisfies the condition $$f(-x) = -f(x)$$ for all x in its domain. Its graph is symmetric about the origin.

The interval of increase for a polynomial is the range of x-values where the function's output is increasing. This can be found by analyzing the derivative.

The interval of decrease for a polynomial is the range of x-values where the function's output is decreasing. This can be determined by finding where the derivative is negative.

To find the x-intercepts of a polynomial, set the polynomial equal to zero and solve for x. The solutions are the x-intercepts.