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ECO 108 Spring 2011 Math Appendix(1)

# ECO 108 Spring 2011 Math Appendix(1) - ECO 108 Math and...

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ECO 108 Math and Graph Appendix Spring 2011 A. Linear Functions A linear function is a function of the form f(x) = y = a + b*x (#) (# is simply a reference for this functional form.) where a and b are constants. The graph of a linear function is a straight line. The slope of a linear function is found by picking any two points on the line, say 1 1 2 2 ( x , y ) and ( x , y ), and determining the ratio of the vertical distance and horizontal distance between the points. slope = 2 1 2 1 y y vertical distance . horizontal distance x x - = - From the function, we can derive information about the line it generates: the slope, the vertical intercept and the horizontal intercept of the line. For the function (#), b = the slope a = the vertical intercept a b - = the horizontal intercept The vertical intercept is the value of y at which the line intersects the vertical axis. The horizontal intercept is the value of x at which the line intersects the horizontal axis. When the line intersects the horizontal axis, the value of the function is 0, i.e., the y-coordinate is zero. Thus, to solve for the horizontal intercept: y = a + b*x and y = 0 a + b*x = 0. So all have to do is solve a + b*x = 0 for x as a function of and b. b*x = – a x = a b - and as stated above, a b - is the horizontal intercept of the line. For example, consider y = 10 – (1/2)*x. If 1 1 2 2 ( x , y ) ( 8, 6 ) and ( x , y ) ( 12, 4), = = the slope of the line can be calculated as 2 1 2 1 y y 4 6 2 x x 12 8 4 - - - = = = - - – 1/2. Moreover, the vertical intercept is 10, and the horizontal intercept is 20. The following diagram shows a graph of this function as well as the points ( 8, 6 ) and ( 12, 4) and a rise – run illustration. Diagram A 0 2 4 6 8 10 12 14 16 18 20 22 0 2 4 6 8 10 12 y x ( 8, 6 ) ( 12, 4 ) +4 - 2 1

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B. Estimation vs. Computation of Slopes: Nonlinear Equation The function (#) from section A generates a straight line , i.e., b is a constant. Unlike linear functions, the graphs generated by nonlinear functions do not have constant slopes, i.e., the slope of a nonlinear curve is not the same at every point on the curve. Thus, the curve generated by a nonlinear function is not a straight line.
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