How To Draw A Inverse Proportion Function Curve?

I'd plot some numbers and sketch the curve by hand.

Or stick some numbers into a spreadsheet and plot from there.

Del

The ONLY - does it excludes a calculator or computer as well?

No.

only by ruler and compasses. pure geometry drawing.

not allow algibra method, from dot to get trace.

Hi,

I don't quite understand which is considered given in this problem. Is it the function formula known? Are some points known and you want to draw a best fit line? If I understand the term "proportional" correctly, then you must be talking about a linear function, i.e. something like y=ax+b with a<0, given that proportionality is inverse.

Now, if you know a and b, or similarly, if you know any two points, you can draw a line, as soon as you can draw the points in the first place! So, the question now is: Can you pinpoint any two points of the given function by simply using a ruler and a compass? If we are talking in geometrical terms, then the general answer is no! Exceptions are the cases where a and b are rationals, or square root of a rational (in which case you can use the pythagorean theorem to get hypotenuses of appropriate lengths, and move them by using the compass).

But you have to be more specific about the problem you are trying to solve. Is it a high-school geometry problem, or what?

equation is simple.

Y=k / X

inverse proportion function. try to draw it curve by only ruler and compasses.

its primary geometry, but not ordinary high school skill.

equation is simple.

Y=k / X

Oops, you are right. I'm not so good in english terminology... Indeed inverse proportion means something like y=k/x (which by the way is not a hyperbola, although it has asymptotes too!)

In order to construct such a curve we need to find points that satisfy x.y = k, that is, corners of boxes with area equal to k. Unfortunately I don't think it is possible to draw boxes of equal area with just a ruler and a compass. Any idea somebody?

The only feasible method that crosses my mind now, is to exploit a property of intersecting circle chords: Say for example that we draw two circle chords AB and CD which cross at point O. Then AO.OB = CO.OD. So if you draw line segments appropriately so that AO.OB = k (e.g. take AO=1 and OB=k), then you can draw as many segments at varying angles passing from O as you like, thus getting pairs of CO's and OD's - or if you like - x and y, which all satisfy x.y = k. Then you transfer those segments to the x-y diagram by using the compass.

To get as many points as possible, you can draw more circles which have AB as their chord. Not so difficult: just chose points on the perpendicular-median (term OK?) of AB to serve as circle centers. The construction is trivial.

One can also use the tangent-secant theorem to get segments which also satisfy x.y = k.

The only tricky point in all the above, is how you draw the line segments appropriately with ruler and compass, if k is neither rational nor root of a rational.

I suppose you could construct a 45 degree downward sloping line and then rig the X and Y scales to fit the function. But I think without using algebra to get the relationship, you couldn't know how to do the scaling.

I suppose that if you consider that the product of X and Y is a constant in an inverse proportion, then a horizontal line representing that constant would also work. But again, you'd need to use algebra to label the axis.

the curve is hyperbola!