[MUSIC] The expression y = f (x) is a general definition. It refers to the fact data mapping is possible. However, it does not tell about the rule or the mapping. This is why at this stage, it is worth illustrating several specific types of function. Each of these functions represents a different rule of mapping. First of all, we have a constant function, which is a function whose range consists of only one element. For example, the function y = f (x) = 8. Which can be also expressed as y = 8 or f (x) = 8. This is a value which remains the same regardless of the value of x. Graphically, such a function will appear as a flat straight line. The constant function can be defined a degenerate case of the so called polynomial functions. These are multi-term functions. Polynomial functions of a single variable x take the general form of y = a0 + a1x + a2x to the power of 2 + up to anx to the power of n. Each term contains a coefficient and the non-negative integer power of the variable x. Notice that we use a0, a1, up to an instead of using a, b, and c. We do this for two reasons. To economize on symbols. And then, we only use the letter a and the subscript is useful in order to identify the location of a particular coefficient in the equation. For example, in a2 is the coefficient of x to the power of 2, and so forth. Depending on the value of the integer n, which indicates the power of x, we have several subclasses of polynomial function. In the case of n = 0, then, y = a0. And in that case, we have a constant function. In the case of n = 1, y = a0 + a1x. In this case, we have a linear function. In the case of n = 2, then, y = a0 + a1x + a2x to the power of 2. And in that case, we have a quadratic function. In the case of n = 3, then, y = a0 + a1x + a2x to the power of 2 + a3x to the power of 3. In that case, we have a cubic function, and so forth. The indicators of the powers of x are called exponents. The highest power of x is often called the degree of the polynomial function. For example, a quadratic function is a second degree polynomial. A cubic function is a third degree polynomial. From a graphical point of view, if we plot a constant function in the Cartesian plane, then, we get the flat line which is symmetric with respect to the y-axis. If we plot a linear function in the Cartesian plane, we will get a straight line. When x = 0, the linear function gives y = a0. And this means that the ordinate pair (0, a0) is on the line. This gives us the so-called y-intercept. The other coefficient a1 measures the slope. The slope is the steepness of our line. This means that a unit increase in x will be translated in an increment in y in the amount of a1. If a1 is higher than 0, then, the slope is positive. This means that the graph is an upward sloping line. If a1 is lower than 0, then, the slope is negative and the line will be downward sloping. If we plot a quadratic function in the Cartesian plane instead, we will get a graph which is called parabola. This is a curve which has a single winger. If a2 is higher than 0, then, the curve is negative. If a2 is lower than 0, then, our curve will be open to the other way. That is, it will represent a valley rather than a hill. If we plot a cubic function in the Cartesian plane, we will get a graph which, in general, has two wingers. These kind of functions are used quite often in the economic and financial models. [MUSIC]
