Precalculus Examples

$\frac{x^{2}}{9} - \frac{{(y - 1)}^{2}}{5} = 1$

Step 1

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 $1$ .

$\frac{x^{2}}{9} - \frac{{(y - 1)}^{2}}{5} = 1$

Step 2

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

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

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, $a$ .

$a = 3$

$b = \sqrt{5}$

$k = 1$

$h = 0$

Step 4

The asymptotes follow the form $y = \pm \frac{b (x - h)}{a} + k$ because this hyperbola opens left and right.

$y = \pm \frac{\sqrt{5}}{3} x + 1$

Step 5

Combine $\frac{\sqrt{5}}{3}$ and $x$ .

$y = \frac{\sqrt{5} x}{3} + 1$

Step 6

Combine $x$ and $\frac{\sqrt{5}}{3}$ .

$y = - \frac{x \sqrt{5}}{3} + 1$

Step 7

This hyperbola has two asymptotes.

$y = \frac{\sqrt{5} x}{3} + 1, y = - \frac{x \sqrt{5}}{3} + 1$

Step 8

The asymptotes are $y = \frac{\sqrt{5} x}{3} + 1$ and $y = - \frac{x \sqrt{5}}{3} + 1$ .

Asymptotes: $y = \frac{\sqrt{5} x}{3} + 1, y = - \frac{x \sqrt{5}}{3} + 1$

Step 9