Hyperbolic Function with Asymmetric Asymptotes

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A hyperbolic function with one horizontal asymptote and another with a slope of 1 is sought, diverging from the standard hyperbola's equal and opposite slopes. The desired function should resemble an exponential for negative x and a linear function for positive x. The discussion suggests that traditional hyperbolas may not accommodate these asymptotic requirements. A proposed alternative involves using the equation xy - y^2 = -k^2, where k influences the y-intercept. The conversation emphasizes the need for a non-standard approach to achieve the specified asymptotic behavior.
hadron23
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Hello,

I wish to find a function similar to,

y^2-x^2=1

but instead of the slope of the asymptotes being +/- 1, I need one of the asymptotes to be of slope 0. That is, I wish to find a hyperbolic function with one horizontal asymptote and the other of slope 1.

Is this possible?
 
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I suppose all hyperbolas have asymptotes of equal slope magnitude but opposite sign, since they can just be thought of as a vertical conic cross section.

The function I am after essentially has to look like exp(x) for negative x and be linear for positive x. Think of an exponential function with a positive linear asymptote. I was hoping a hyperbolic function may be able to do this, but I don't believe it will. Any ideas?
 
Do you know how to write the equation of a hyperbola aligned with the xy axes, centered at the origin whose asymptotes are lines through the origin with slope ±pi/8 and with intercepts on the y axis? If so, then rotate the graph by pi/8 and you will have your equation. Besides the x2 and y2 term you will also have an xy term in your answer.
 
If you are still here, think about this much simpler equation:

xy - y^2 = -k^2

The value of k will determine the y intercept.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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