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

• Dogtanian
In summary, the conversation revolved around finding an equation for a graph with asymptotes at y=x and y=-x. After some discussion, it was suggested to rotate the graph of y=1/x to achieve the desired result. Later, it was mentioned that the hyperbola x² - y² = 1 can be rotated to fill all four quadrants, and this can be simplified to (x² - y²)² = 1. However, it was also noted that this curve cannot be described by a function of 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 :)

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

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

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

Dogtanian said:
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.

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.

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

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

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.

...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.

## 1. What is an asymptote?

An asymptote is a line that a curve approaches but never touches. It is often used to describe the behavior of a function as the input values get larger or smaller.

## 2. How do you find the asymptotes of a function?

To find the asymptotes of a function, you can set the denominator of the function equal to zero and solve for the input values that make the denominator zero. These input values are the vertical asymptotes. For horizontal asymptotes, you can look at the highest degree terms in the numerator and denominator of the function and compare their coefficients.

## 3. What are the equations for the lines y=x and y=-x?

The equation for the line y=x is y = x. This line has a slope of 1 and passes through the origin. The equation for the line y=-x is y = -x. This line has a slope of -1 and also passes through the origin.

## 4. How do you know if a function has an asymptote at y=x or y=-x?

A function will have an asymptote at y=x if the highest degree term in the numerator and the highest degree term in the denominator have the same coefficient. Similarly, a function will have an asymptote at y=-x if the highest degree term in the numerator has a coefficient that is the negative of the highest degree term in the denominator.

## 5. Can a function have both y=x and y=-x as asymptotes?

Yes, a function can have both y=x and y=-x as asymptotes. This means that the function is approaching both of these lines as the input values get larger or smaller. This can occur when the highest degree terms in the numerator and denominator have the same absolute value but different signs.

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