Hyperbolic Function with Asymmetric Asymptotes

[hadron23]

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?

 

[hadron23]

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?

 

[LCKurtz]

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.

 

[LCKurtz]

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.

 

[CellCoree]

Yes, it is possible to find a hyperbolic function with asymmetric asymptotes. One example is the hyperbolic cosine function, which has a horizontal asymptote at y=1 and a slanted asymptote with a slope of 1. This function is defined as cosh(x) = (e^x + e^-x)/2. Another example is the hyperbolic tangent function, which has a horizontal asymptote at y=1 and a slanted asymptote with a slope of 1. This function is defined as tanh(x) = (e^x - e^-x)/(e^x + e^-x). These functions have been extensively studied and have many applications in mathematics and physics. I would recommend further research and exploration of these functions to better understand their properties and behavior.