Polynomials are perhaps the most common functions encountered in everyday problems. Sums and differences of different powers of the independent variable are a function of the dependant variable. For example, in y = 3x^3 - 2x^2 + x - 5, y is a function. The degree -- the largest exponent -- of the polynomial tells a lot about the shape of the function's graph. The graph always goes off to infinity -- increases without limit -- as x increases in a positive direction. Odd degree polynomials go to negative infinity as x decreases in a negative direction. Even degree polynomials go to positive infinity. The graph may change directions several times in between, but the maximum number of direction changes is the degree minus one.
Rational functions are those where there is division -- a fraction. The numerator and denominator of the fraction may be polynomial such as y = (x^2 + 3) / (x - 1). The interesting thing about these functions is what happens when the denominator equals zero. Division by zero is undefined, but division by very small numbers produces very large values for the function. When you graph a rational function, you will have vertical asymptotes -- lines that the graph approaches closer and closer, but never reaches -- at all the places where the denominator goes to zero.
The two most studied and most common trigonometric functions -- sine and cosine -- are periodic functions that are bounded by the values -1 and +1, and the graphs have no asymptotes. The sine and cosine are identical except one is 90 degrees out of phase with the other. All the other trigonometric functions have vertical asymptotes every 180 degrees. You can predict where the asymptotes will be if you remember the identities that express the other functions in terms of sine and cosine. For example, the cotangent is equal to the cosine over the sine. It has a sine in the denominator, so the cotangent will have vertical asymptotes at the places where the sine equals zero: 0 degrees, 180 degrees, etc.
The conic sections include circles, ellipses, parabolas and hyperbolas. The Greeks called these conic sections because they arise when planes intersect cones. Only the hyperbolas have asymptotes, but they all contain foci -- points that are essential in defining the curves. The simplest of these is the circle -- defined as all the points that are the same distance from the center -- a single focus. Parabolas are defined as all the points that are equidistant from a focal point and a fixed line.