The end behavior of a polynomial function is determined by its degree and the sign of its leading coefficient, that is, the coefficient of the highest degree monomial. As x approaches to infinity, a positive leading coefficient makes the graph extend upward, and a negative one makes the graph extend downward. For even-degree polynomials, the left-end behaves the same way as the right-end. For odd-degree polynomials, they have opposite behavior.
Four graphs of polynomial functions are shown in the image.
From the end behavior of the graphs, we can determine whether they have a positive or negative leading coefficient and if the function is of even or odd degree. By looking at the right-end of the graphs, we can find the sign of the leading coefficient. If it extends upward, it's positive, and if it extends downward, it's negative.
Using some of the characteristics of a polynomial function, such as its zeros and end behavior, a rough sketch of its graph can be drawn. For instance, the graph of a polynomial function with the following characteristics can be sketched.
Its zeros are x=-2, x=1 and x=4. As x approaches positive infinity, the graph of the function extends upward, and as x approaches negative infinity, it extends downward.
Begin by plotting each zero in a coordinate plane. Here, the coordinates of the zeros are (-2,0), (1,0) and (4,0).
Next, mark the graph's end behavior in the coordinate plane. Note that the ends must be sketched to the left of the leftmost zero and to the right of the rightmost zero, or the graph will not represent a function. The example's right-end extends upward and its left-end extends downward.
Lastly, the points and the ends can be connected by a smooth curve. Note that the curve must change from increasing to decreasing or vice versa at least once between two zeros.
The graph above fulfills the criteria for the function. However, there are an infinite number of other polynomials that also do so. Another example is sketched below.
Thus, the information given was not enough to specify a single function.