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the slope of a linear function
where a and b are known constants. The value a is called the intercept and the value b is called the slope. Mathematically, the intercept is the value of y when x = 0. This expression is said to be linear in x and y, where the corresponding graph is represented by a straight line. In the total revenue example just given, a = 0 and b = 18.
THE SLOPE OF A LINEAR FUNCTION In Equation (2.5), the parameter b is called the slope, which obtained by dividing the change in the value of the dependent variable (the “rise”) by the change in the value of the independent variable (the “run”) as we move between two coordinate points. The value of the slope may be calculated by using the equation Dy Dx y2 - y1 f (x 2 ) - f (x1 ) = = x 2 - x1 x 2 - x1
Slope =
(2.6)
where the symbol D denotes change. In terms of our total revenue example, consider the quantity—total revenue combinations (Q1, TR1) = (1, 18) and (Q2, TR2) = (3, 54). We observe that these coordinates lie on the straight line generated by Equation (2.4). In fact, because Equation (2.4) generates a straight line, any two coordinate points along the function will suffice when one is calculating the slope. In this case, a measure of the slope is given by the expression DTR DQ TR2 - TR1 f (Q2 ) - f (Q1 ) = = Q2 - Q1 Q2 - Q1
b=
(2.7)
After substituting, we obtain b=
54 - 18 36 = = 18 3-1 2
which, in this case, is the price of the product. Suppose that we already know the value of the slope. It can easily be demonstrated that the original linear function may be recovered given any single coordinate along the function. The general solution values for Equation (2.5) may be written as b( x2 - x1 ) = ( y2 - y1 )
