5. Linear and Exponential Equations

    The equation for a straight line is "y = mx + b", in which "m" and "b" are constants.  In the example (Linear Equation), "m" has been set to 2 and "b" to 4.  "b" is called the y-intercept, because when x = 0 (which is the location of the y-axis), y = "b".  "m" is called the slope of the line, and it can be calculated by taking any two points and dividing the difference in y values by the difference in x values [i.e., m = (y2 – y1) / (x2 – x1)].  Notice that this is the same as calculating a rate.  In the example, if we use the last two data points in the table, then "m" = (22 – 16) / (9 – 6) = 6 / 3 = 2.

    A simple exponential equation is "y = axb", in which "a" and "b" are constants.  Let’s see how different values of "b" changes the graph by setting "a" = 1 (Exponential Equations).  If "b" = 1, then this is simply a straight line with a y-intercept of zero (#1 in the example) and a slope equal to "a" (in this case, 1).  If "b" is greater than 1, then the graph curves upward (#2 in the example).  If "b" is less than 1 but greater than 0, then the graph flattens out (#3 in the example).  If you are curious about the effects of negative values for either "a" or "b", just make your own graphs and see what happens.

    When the basal metabolic rate (MR) was measured for a number of different species and graphed as a function of each species’ mass (M), it was found that the graph flattens out, as in #3 in the Exponential Equations example.  Hence, MR = aMb.  This curve shows, on the one hand, that larger animals have a higher MR than smaller animals, because MR increases with increasing mass.  On the other hand, for their size larger animals have a lower mass-specific MR than small animals, because the curve stays below the "b" = 1 line.

    How can the value of "b" be determined?  The first step is a log transformation of our exponential equation.  That is, if "y = axb", then "log(y) = log(axb) = log(a) + b*log(x)".  You should recognize this as the equation for a straight line, in which the y-intercept is "log(a)" and the slope is "b".  In the example (Exponential Equations), the log values for x and y were calculated and graphed; all three data sets now form straight lines.  Using the last two data points in the table for #3, then the slope is (0.716 – 0.678) / (0.954 – 0.903) = 0.745, which is close to the value of "b".

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