SOLUTION: y^2-x^2/15=1 Want to know the focus,vertices and the asymptote

Question 621020: y^2-x^2/15=1
Want to know the focus,vertices and the asymptote
Answer by KMST(5328) (Show Source): 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).
RELATED QUESTIONS
What is the vertices, foci, and slope of the asymptote for the hyperbola whose equation... (answered by venugopalramana)

Equation of a hyberbola (x-3)^2 over 4 - (y+2)^2 over 9 = 1 How do I find the... (answered by lwsshak3)

I need to know the slant asymptote and the X-Intercept, and Y-Intercept of this problem... (answered by robertb)

What is the vertices, foci, and slope of the asymptote for the hyperbola whose equation... (answered by venugopalramana)

it says, "Identify the curve and find the characteristics listed. then sketch the curve." (answered by lwsshak3)

Find the first TWO non-negative asymptotes and the first negative asymptote of the graph... (answered by jsmallt9)

What is the vertices, foci, and slope of the asymptote for the hyperbola whose equation... (answered by venugopalramana)

1. Identify the focus, directrix, and axis of symmetry of the parabola a. y=... (answered by ewatrrr)

How to identify and graph the asymptote and y intecept for graphing exponential growth. (answered by drk)