Precalculus Examples

$16 y^{2} - x^{2} - 16 = 0$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Add $16$ to both sides of the equation.

$16 y^{2} - x^{2} = 16$

Step 1.2

Divide each term by $16$ to make the right side equal to one.

$\frac{16 y^{2}}{16} - \frac{x^{2}}{16} = \frac{16}{16}$

Step 1.3

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$y^{2} - \frac{x^{2}}{16} = 1$

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.

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

Step 3

Match the values in this hyperbola to those of the standard form. The variable $h$ represents the x-offset from the origin, $k$ represents the y-offset from origin, $a$ .

$a = 1$

$b = 4$

$k = 0$

$h = 0$

Step 4

The center of a hyperbola follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(0, 0)$

Step 5