what does the graph of x^2 - y^2 = 5 look like?

Question

how do i determine the points of this graph?

what is the asymptote?

Answer 1

It's a hyperbola with its principal axis the x-axis. It has asymptotes at +/- 45 degrees, that is along x = +/- y, and is centred on the origin O at (0, 0). It "opens" in the +/- x directions, and its closest approaches to O are at the points (x, y) = (+/- 5^(1/2), 0).

Live long and prosper.

LATER: I hope you enjoyed trying to follow cherrypassion's advice to use x-values in the integer range from - 2 to + 2, in order to plot the function. If you did, you will no doubt have found that all the associated y-values were in fact imaginary! That is so of course, because, as I pointed out, the closest the hyperbola ever comes to the origin is at +/- 5^(1/2) = 2.2361... on the x-axis. (That value of x corresponds to y = 0.)

If you DO want to plot this hyperbola yourself, there are naturally REAL values of x corresponding to ALL integer y-values from, say, - 4 to + 4. Finding the corresponding x's and plotting the resulting (x. y) pairs will give you a good idea of the general form of the curve.

As to your second question under "additional details" : Since I told you what the MATHEMATICAL FORM of the asymptotes is, in my answer's 2nd sentence, I take it that your question "what is the asymptote?" is really "what does the term asymptote itself mean?"

In its simplest form, an asymptote is a straight line to which the graph of a function tends as x and/or y tend to +/- infinity, without ever quite reaching it. Thus, the plot of xy = c^2 (which is also a hyperbola) has 4 different asymptotes. These are simply the limits of the x- and y-axes as they themselves tend to infinity. That is because, for example, the LARGER x gets, the smaller y BECOMES, without limit. That means that as x --> + ∞, y remains positive but --> 0, without ever quite getting there. Similar things are true at the three "other ends" of the coordinate axes. (Note that if c^2 = 5 / 2, this second hyperbola is simply the original hyperbola you asked us about, rotated through 45 degrees. It exists only in the first and third quadrants.)

At a more sophisticated level, a more general idea is that of "asymptotic behaviour." It may be that a complicated function tends to a simpler analytical (non-straight-line) form as x and/or y tends to +/- infinity. In that case, one says that the original function behaves like that simpler function "asymptotically." That is then a generalization of it tending to behave like certain straight lines; it may now be tending to behave like certain well-known or otherwise well specified curves.

You might be interested in the asymptotic form (BEYOND the leading term) of the original function you asked about:

Since x^2 = y^2 + 5 = y^2 (1 + 5 / y^2), the asymptotic form is

x = +/- y [1 + 5 /(2 y^2) + O(1 / y^4)] = +/- [y + 5 / (2y) + O(1 / y^3)].

So that shows you exactly HOW the asymptotic forms x = +/- y are approached as y, and thus x also, tend to +/- infinity.

Answer 2

well, to begin with, you can choose a range of values for x (say -2,-1,0,1,2).

put in these values one by one and get the corresponding values of y.

once that is done, you'll have pairs of x,y values.

on a sheet of graph paper, plot these points and join them to draw the graph of the equation.

Answer 3

This is a hyperbola that opens east-west. center at origin

Answer 4

This is a hyperbola