To "flip" a function across the x-axis, simply negate the function, e.g., becomes .
This is for a practical application, that is the manufacture of a new horn for a wind up gramaphone..........sometimes called "exponential horns"
I expect many here will be familiar with the development of a cone on a flat surface, with a constant taper it is just a straight line at either end but with a gramaphone type horn, you require a curve on one side and its mirror image on the other.....of course, I could just plot a suitable curve, cut it out on card and flip it over to draw the other side, but if I were to give the design to a cnc machine, to be punched out of a metal sheet, I would require the equation of the curve and its mirror image, I have a graph drawing calculator but have no idea how to change the equation to produce the mirror image curve.
Any advice on best curve and how to alter it as above would be most gratefully recieved,
Hello, thanks, it did occur to me much later on to try that, on my calculator I have managed to plot something that looks like the right sort of shape with y=e(x-2.178) and y=1/(e(x-7.178)) but I don't know if that is the best function to use, the aim being to make a short tube capable of resonating at low frequencies, to reproduce the low notes that musical instruments can produce, for example, a Bflat Euphoniam has to be over eight feet in length to produce that bottom Bflat.....you obviously can't easily put such a long horn on a wind up gramophone, and even if you did, there wouldn't be enough energy in the needle and diaphram to move that amount of air.....so, it becomes a question of what is the best shape for a short horn....there probably isn't an answer to that question but there may be some theory that would suggest an optimum curve and length of horn
Well, I do play trombones (an E-flat alto and a B-flat/F tenor) and yes I think if I recall correctly the B-flat horn is about 9 ft. long. As far though as the optimal shape for a short horn, I don't know.