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

Jun 23, 2018

Vertices: $\left(\pm 2 , 0\right)$
Foci: $\left(\pm 2 \sqrt{2} , 0\right)$
Asymptotes: $y = \pm x$

#### 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

${\left(x - h\right)}^{2} / {a}^{2} - {\left(y - k\right)}^{2} / {b}^{2} = 1$

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:

${a}^{2} + {b}^{2} = {c}^{2}$
$4 + 4 = {c}^{2}$
$8 = {c}^{2}$
$2 \sqrt{2} = c$

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: $\left(\pm 2 , 0\right)$
Foci: $\left(\pm 2 \sqrt{2} , 0\right)$

Finally, let's get the equations of the hyperbola's asymptotes. The equations are $y = \pm \frac{b}{a} x$ for a horizontal hyperbola at $\left(0 , 0\right)$. Therefore, our equations are $y = \pm x$.