Asymptotes as the lines y=x an y=-x

[Dogtanian]

Hi there.

I'm a maths teacher and today was having a discussion with my Head of Department about asymptotes (as you do!)

She was me if I could think on an equation of a graph(s) which has an asymptotes at the line y=x and another at y=-x.

Thinking about it, neither of us could come up with anything. It's really bugging me now. I ain't too sure if there is such a function...though it seems silly there not being.

The only way I managed to get a graph like what I wanted was to rotate the graph of y=1/x by various angles.

So can such a graph exist? Is it possible to transform equations of graphs via rotations (something I can't ever remember doing)?

Hope I can come up with an answer by tomorrow :)

 

[Matthew Rodman]

How about

$$y = x \tanh{x}$$ ?

as

$$x \tanh{x} = x \frac{e^x - e^{-x}}{e^x + e^{-x}}$$

therefore, for large positive x,

$$x \tanh{x} \approx x \frac{e^x}{e^x} = x$$

and for large negative x

$$x \tanh{x} \approx x (\frac{-e^{-x}}{e^{-x}}) = -x$$

 

[arildno]

Hi there.

I'm a maths teacher and today was having a discussion with my Head of Department about asymptotes (as you do!)

She was me if I could think on an equation of a graph(s) which has an asymptotes at the line y=x and another at y=-x.

Thinking about it, neither of us could come up with anything. It's really bugging me now. I ain't too sure if there is such a function...though it seems silly there not being.

The only way I managed to get a graph like what I wanted was to rotate the graph of y=1/x by various angles.

So can such a graph exist? Is it possible to transform equations of graphs via rotations (something I can't ever remember doing)?

Hope I can come up with an answer by tomorrow :)

You certainly can rotate curves!

For example, the curve described by the equation:
$$x^{2}-y^{2}=1$$
has asymptotes y=x and y=-x.

Note, however, that in this case, there exists no function of x, by which the y-coordinates of the curve could be computed, and the curve in question cannot be regarded as the graph of some function, i.e, the set of points describable as (x,f(x)), where f is some function.

 

[robphy]

Doesn't one branch of that hyperbola still have those lines as asymptotes?
In a physics context, the worldline of a uniformly accelerated observer [which can be regarded as function (t,x(t)), where x(t)=sqrt(1+t^2)] is asymptotic to a light cone.

 

[Dogtanian]

Thanks for the help guys :D

My head of department was so sure you could each of the 4 sections between the said asymptotes filled with a curve.

Using what the first two posts said, I drew the graphs of

y = sqrt(1+x^2)
y = -sqrt(1+x^2)
x = sqrt(1+y^2)
x = -sqrt(1+y^2)

to get what I needed (at least I think that is what I i if I remember correctly back to last night...)

But I only went and forgot all about this today until just now...so I never i speak to my HofD about it...never mind :D

 

[Hurkyl]

You can combine the four curves to get one that fills the whole thing.

We already know that

x² - y² = 1

fills in two sections, and

y² - x² = 1

fills in the other two.

We can combine them into one equation:

(x² - y² - 1) (y² - x² - 1) = 0

which will fill in all four quadrants.

 

[robphy]

...which can be simplified to read (x² - y²)² = 1.
Of course, the easy way to see this is to note that
y² - x² = 1 is the same as x² - y² = -1.