How do I sketch a polynomial function?

Before you start it is best to gather as much information as you can about the function. The most important things to find are any roots - as these will tell where the graph should cross the axis - and the turning (max/min) points of the graph. Roots can be found by solving/factorizing the polynomial and the turning points should be found by differentiating the function and then finding the roots of the derivative. Some useful information can be gleaned from the polynomial's order (the highest power on our variable, usually x) and from the factorized form of the polynomial too. The order of the polynomial tells you how many turning points there should be if the order is zero or one there are no turning points as we have either a flat or slanted line, if the order is two we have one turning point if it is 3 two turning points and so on. The number of times a term appears in the factorization of the polynomial also helps us out if a term appears only once the graph only crosses at the point if a term appears twice then the point is a turning point and if it appears three times we have a point of inflection. Once you've found these points plot them and label them on your graph. The last stage is to join up the points with a smooth curve. To tell if the graph to the left of our first point comes from positive or negative infinity we look at the sign and the order of our highest order term. If the order is even and the sign is negative then it comes from negative infinity and if positive, positive infinity. If the order is odd then a positive sign means negative infinity and a negative sign positive infinity. To find where the graph goes after the last of our points we just extend the graph in the direction that our last point was directing.