You can
put this solution on YOUR website!For hyperbolas, we like the standard form of the equation, the one that shows a difference of squares equal to 1.
It may look like
or
.
That form of the equation shows you all the numbers you need to know to figure out the he foci, vertices and the asymptotes. (All you need is the a and b numbers).
is the equation (in standard form) of a hyperbola centered at the origin, so this is an easy problem.
We know this one is centered at the origin because there is just an
and a
, with nothing added or subtracted before squaring.
Because of that simplicity, it is easy to see that changing x to -x gives you the same equation, meaning that the graph is symmetrical with respect to the y-axis. The same can be said of changing y to -y, and the symmetry with respect to the x-axis.
For y=0 we would have a negative number equal to 1
and that cannot be.
So, we can see that the graph does not touch the x-axis, where y=0. (In fact the graph does not even want to get close to the x-axis)
On the other hand, y cannot be zero, but x can be zero.
When
, you see that
, meaning
or
, so the graph goes through the points (0,1) and (0,-1).
For all other points,
so
and
, meaning that all the other points are even farther away from the x-axis, where y=0.
The closest that the hyperbola comes to the x-axis is the points (0,1) and (0,-1) , which are the vertices.
As x (and y) grow larger in absolute value,
and
grow larger, and the graph gets closer to the asymptotes.
A little algebra transforms the equation into one that gives us the equations of the asymptotes:
-->
-->
-->
-->
As
grows larger,
grows smaller, and the graph gets closer to the graph for
which is the graph for the lines
<-->
and
<-->
.
Those lines are the asymptotes.
Because teachers do not like to see square roots in denominators, we may have to write them as
and
.
THE FOCI:
There foci are at a distance
from the center of the hyperbola, and the number
is related to the numbers
and
in the standard form of the equation by a formula that can be derived using the Pythagorean theorem. It is
In this case, your
and
are 1 and 15, so
-->
-->
.
The center was (0,0) (the origin).
The vertices were ((0,-1) and (0,1), on the y-axis.
The foci are on the same line, but at distance 4 from the center/origin, at
(0,-4) and (0,4).
