Question

How do you find the coordinates of the vertices, foci, and the equation of the asymptotes for the hyperbola `x^2-y^2=4`?

Answer 1

Vertices: `(+-2, 0)`
Foci: `(+-2sqrt2, 0)`
Asymptotes: `y=+-x`

Explanation:

First, we need to have our equation in standard form:

`x^2/4-y^2/4=1`

That's better! To get to our answer, we need to know that our hyperbola is a horizontal hyperbola, which means that the vertices and foci will be have the same y-coordinate as the center.

Next, we need to know our the values of `a`, `b`, and `c`. These help us find out the dimensions of our graph. The formula for a horizontal hyperbola is

`(x-h)^2/a^2-(y-k)^2/b^2=1`

In our case, the `h` and `k` values don't exist, so our center is at the origin. `a^2` and `b^2 =4`, so both `a` and `b = 2`. To find `c`, we will use an equation you've probably seen before:

`a^2+b^2=c^2`
`4+4=c^2`
`8=c^2`
`2sqrt2=c`

To find our vertices and foci, we move out a certain amount of units in both directions: `a` for the vertices and `c` for the foci.

Vertices: `(+-2, 0)`
Foci: `(+-2sqrt2, 0)`

Finally, let's get the equations of the hyperbola's asymptotes. The equations are `y=+-b/a x` for a horizontal hyperbola at `(0,0)`. Therefore, our equations are `y=+-x`.