or more generally:

Let's simplify our lives and consider an example centered at the origin:

Then they divide the world into those hyperbolas that open up and down versus those that open up left and right.

By looking at the equation, I think we can agree that if x

^{2}equals 0 we've got a problem, since no value of y can satisfy the equation, and in fact x must not be less than 1. Hence, we can already predict that this system opens left and right, as the plot shows (

`a`^{2} = b^{2} = 2

):The ALEKS review goes on to explain that for this type of equation, the vertices of the hyperbola (points of closest approach to the origin at

`h,k`

) are equal to and the asymptotes have slopes equal to

This all works great. The problem I had was that the simplest parabola I can think of is:

which doesn't fit the system.

The answer to my confusion is that any parabola may be rotated around its origin in the xy-plane. In 2 selected orientations its equation will have only

`x`^{2}

and `y`^{2}

terms (when the vertices are on the x or the y-axis), whereas in 2 other selected orientations it may have only `xy`

terms (when the vertices are on `y= +/- x`

. The rest of the time it contains both.This is explained in an excerpt from Stewart's Calculus I found here.

Consider a point

`P`

in the plane with coordinates `x,y`

and distance from the origin `r`

. Now, let's establish a new coordinate system `u,v`

which is rotated counter-clockwise by the angle `θ`

. A vector from the origin through `P`

is rotated an angle `φ`

with respect to the u axis and `φ + θ`

with respect to the x-axis. Here's a screenshot from the pdf:(Note, he uses capital X and Y for the second set of coordinates).

In the

`u,v`

system, the point `P`

has coordinates In the

`x,y`

system the coordinates are:We remember (my post here) that:

Hence:

Now, consider the hyperbola:

Rewrite this in terms of

`u,v`

and multiply, giving 4 termsIf

`θ`

equals `π/4`

, then The two middle terms drop out and leave us with:

If the middle terms don't cancel, we're left with a mixture including some fraction of

u

^{2}/2, v

^{2}/2, and u times v.

Note: we rotated the coordinate system counter-clockwise, which has the effect of rotating the plot clockwise, when the coordinate system is viewed in standard orientation.