Important note as the above example shows we can make

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IMPORTANT NOTE : As the above example shows, we can make a simple little lemonade stand sound fairly complicated once we translate it into “mathe- matese.” And in fact, in this case you could have done the calculations more easily in your head (although in fact what you would have been doing was just what the function says you should have done). With more complicated relations, however, M1-2 MATH MODULE 1: ±UNCTIONS, GRAPHS, AND THE COORDINATE SYSTEM y y = f ( x ) 1 0 1 2 2 = 0 0 FIGURE M.1-1
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where the solution may involve many steps, “doing it in your head” increases the possibility of error, and may be (practically speaking) nearly impossible. One of the tricks in using mathematics, rather than being used or pushed around by it, is to be able to translate it back into common sense terms, whenever required. Functional notation is a remarkable communication device. With just a few symbols, it can provide a summary of the relation between an infnite number of values of one variable and the corresponding values taken on by another variable. We simply need to remember the function, and then just plug in the specifc values of one variable that we are interested in and perform the operations on them that the function tells us to, to ±nd the corresponding specifc values for the other variable. Functional notation thus saves time, economizes on memory, and frees up our mind for real thinking. It can even some- times suggest new ideas or connections and analogies that we hadn’t seen before, and thus improve our thinking. We now briefly review coordinates and graphs, and then look at some specifc aspects of functions as they are used in economics. 1.2 GRAPHS AND COORDINATES With a few exceptions (see Appendices 3 and 9 in the text, where you will ±nd some 3-dimensional graphs), economists typically use 2-dimensional graphs to depict function- al relationships. Graphs have the advantage of giving us much qualitative information regarding a functional relationship in visual form, so that we can grasp it “at a glance.” Graphs also have some disadvantages: they are less precise than algebraic representa- tions of functions, and on occasion they can be misleading. Yet on balance they are extremely valuable aids to economic understanding. MATH MODULE 1: FUNCTIONS, GRAPHS, AND THE COORDINATE SYSTEM M1-3 y x xx (0, 12) (0, 8) (4,0) (8,0) (12,0) (0, 4) 0 A (4, 8) B (8, 4) (0, –) (0, +) (–, +) (+, +) (–, –) (a) (b) (+, –) (–, 0) (+, 0) H C FIGURE M.1-2
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Figure M.1-2(a) contains a graph of a simple linear (straight-line) function, y = 12 – x , which illustrates a number of the important conventions we use. The x and y axes are at right angles (or “ orthogonal ”) to each other. We designate the horizontal axis as the x axis and the vertical axis as the y axis. The x axis is de±ned by the equation y = 0, and the y axis by the equation x = 0. Each point P 0 in the ( x, y ) plane may be characterized by a pair of numbers ( x 0 , y 0 ), with the x value always coming ±rst. Point A has the coordi- nates ( x A , y A ) = (4, 8), because it is 4 units to the right (“east”) of the vertical axis and 8 units above (“north of”) the horizontal axis. Analogously, point
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