A) How is the end behavior of a polynomial graph determined? Completely describe all 4 cases and express the behavior with the proper symbols. Consider the 2 cases of a polynomial in standard form and a polynomial in factored form.

B) How are the x-intercepts of a polynomial found and expressed? What could the graph do at the x-intercepts and how is that determined? Consider the 2 cases of a polynomial in standard form and a polynomial in factored form.

C) How is the y-intercept found and expressed? Consider the 2 cases of a polynomial in standard form and a polynomial in factored form.

A) Let the leading term of the polynomial be ax^n. Then... ..if n is even, then the end behavior is the same on both ends; the graph on both ends goes to positive infinity if a>0 or to negative infinity if a<0 ..if n is odd, the end behavior is opposite on the two ends; if a>0 then the graph goes to positive infinity as x goes to infinity and goes to negative infinity as x goes to negative infinity; if a<0 then the graph goes to negative infinity as x goes to infinity and goes to positive infinity as x goes to negative infinity. If the polynomial is in factored form, then it is easy to determine the leading coefficient; once you have that, the analysis is as above.

B) The x-intercept(s) are where the function value is 0. It is easy to see where the zeros are if the polynomial is in factored form; if the polynomial is in the standard form, then you need to find the factored form or use some tool like a graphing calculator to find the zeros. Assuming then that you have a polynomial in factored form, one of two things can happen with the graph at the root. ..If the root is an odd multiplicity, then the graph crosses the x-axis from positive to negative, or vice versa. That is because, as you "walk" along the x-axis from one side of the root the other, an odd number of factors change sign, resulting in a sign change for the polynomial. ..If the root is an even multiplicity, then an even number of factors change sign from one side of the root to the other, resulting in no sign change for the polynomial. So the graph just touches the x-axis but stays on the same side of the x-axis on both sides of the root.

C) The y-intercept is where the x value is 0. It should be obvious that if the polynomial is in standard form, then the y-intercept is just the constant term (or 0, if there is no constant term). If the polynomial is in factored form, the y-intercept is just the product of all the constant terms in the factors.

Mathematics