I believe that mpg and fuel cost savings are a bit confusing because they involve a *hyperbolic function*. But we are stuck with mpg by tradition; perhaps the attached chart will help simplify things. It assumes 12,000 miles driven per year, and gasoline costs $2.50/gallon. One implication from the hyperbola is that the dollar savings when increasing from 20 to 30 mpg is equal to that from 30 to 60 mpg.

Attached Files

hyperbolic.xls (16.0 KB, 15 views)

first, i'd go with "quadratic" rather than "hyperbolic."

second, by taking your graph and changing both the vertical and horizontal scales to logarithmic, the graph turns into... (drum roll, please...)

a straight line. much easier to extrapolate and interpolate, bytheway.
whenever you're faced with a curved line on a graph, it's not graphed on the correct axes. change one or both to log scale until the line is straight. you can't interpolate or extrapolate a curved line on a piece of paper or on a terminal screen.

+af
Northwest Raleigh, NC, USA
'04 Silver opt.BC/9; 24,000+ miles so far
http://www.plusaf.com/prius/prius.htm
http://www.plusaf.com/coast/coast1.htm
My Prius without the trailer:

My Prius with the trailer:

I respectfully disagree, plusaf. This curve is a section of a rectangular hyperbola of the function X*Y = 30,000. Just in the positive quadrant of the Cartesian plane, of course. 30,000 being the product of 12,000 miles per year and $2.50/gallon. I agree that a logarithmic transformation of both axes would linearize this function, but it will not make our lives any easier.

If one accepts the conditions of 12,000 miles per year and $2.50/gallon, then the section drawn ought to serve the needs of most. On this curve, annual $ fuel costs (Y) will be 30,000 divided by miles per gallon (X).

Or, to change the conditions, just make a new hyperbola. For example, if 15,000 miles/yr and $2.75/gallon, X*Y = 41,250. Y = 41,500/X.

Quadratics and trigonometry would be required for a full treatment of hyperbolic functions, but thankfully this is the simplest example, and we can set all that aside here. I honestly intended this to be a simple fuel-cost calculator, and hope that despite my response here, it still can be used as one.

DAS

I would imagine that the resale value of a hybrid would primarily rely on two factors: Availability of NEW hybrids; Price of gasoline.

-- If new hybrids are plentiful and gas is at a" low" point, a used hybrid would probably depreciate at the same rate (or maybe a little faster) than a regular car.
-- If new hybrids are plentiful and gas is at a "high" point, a used hybrid might depreciate a little more slowly than a regular car.
-- If new hybrids are in short supply and gas is at a "high" point, then a cosmetically perfect used hybrid probably could command a near-new price (because new hybrids will be selling at above-new prices).

At least that's what I think...

This thread is getting mathematically astray, but at least I didn't start it down that path. I did think about posting something like plusaf's first post in this thread though.

What?

And then try to explain to others what the graph really means. People are barely comfortable with graphs and you expect them to interpret things on a logarithmic scale? It's especially hard if you only change one axis.

All plusaf is saying about picking the 'correct axes' to graph something on is that most things people want to graph can be expressed as a direct relationship between two or more variables. Sometimes the variable is log y or y^2 or sin 2y, -here it is 1/y, as the OP stated- so for us techies it's often easier to worry about calculating everything in terms of 1/y, but for the non-techies who can figure out that y is the variable and that's about it, having axes that are non-linear can be really messy. This is especially true because it's one way that people lie with statistics- if you use a log scale or set the origin at 60% on one axis rather than 0 (to over-emphasize small differences), it is much easier to make people misunderstand and misinterpret data that is otherwise correct. So that's the danger in changing the axes- it can be useful to people who follow what you're doing, or it can be misleading to people who cannot.

I think that Tochatihu's graph was just fine as is, and that many laypeople could understand it, which is probably the most important thing. Yay for data!