The roots of an equation can be seen clearly in the equation's graph. For a polynomial, the degree -- the value of the largest exponent -- gives the maximum number of roots. Each time the graph of the equation crosses the x-axis -- places where y is zero -- represents a root. Only real valued roots are indicated by x-axis crossings, so counting the x-axis crossings reveals how many real roots and how many complex roots exist. If the graph of a trinomial crosses the x-axis one time, there are one real and two complex roots. If the graph of another trinomial crosses the x-axis three times, there are three real roots and no complex roots.
Extrema are local minimums and maximums -- places where the curve changes direction, making a small hill or valley in the graph. These numbers are very important to people who interpret graphs when they want to do things like make a tin can with the minimum amount of tin when the can must have a given volume, or the minimum amount of fence to enclose a specific area against an unusually shaped building. The degree of a polynomial -- the largest exponent -- tells how many times the curve will change direction. The extrema will always be equal to the degree minus one.
Asymptotes are lines to which curve comes closer and closer but never reaches. An example is the equation y = 1/(x^2 -1). There are two vertical asymptotes in this graph, at x = -1 and x = 1. As x approaches either -1 or +1, the value goes off to either positive or negative infinity -- approaching the asymptotes more and more closely but never really touching them. Notice that, in the equation, when x = -1 or x = 1, y is undefined.
Exponential curves can be symmetric about their axes in two different ways: even symmetry and odd symmetry. With even symmetry, the curve looks like it is a mirror reflection around the y-axis. A feature close to the y-axis on the left will be close to the axis on the right; in mathematical terms, even symmetry is when f(x) = f(-x). In odd symmetry, the curve is flipped around both the X and Y axes, so the image in the first quadrant is duplicated in the third quadrant. Mathematically, odd symmetry is when f(x) = - f(-x).