Precalculus Examples

$16 y^{2} - 4 x^{2} - 64 = 0$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Add $64$ to both sides of the equation.

$16 y^{2} - 4 x^{2} = 64$

Step 1.2

Divide each term by $64$ to make the right side equal to one.

$\frac{16 y^{2}}{64} - \frac{4 x^{2}}{64} = \frac{64}{64}$

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 $1$ .

$\frac{y^{2}}{4} - \frac{x^{2}}{16} = 1$

Step 2

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

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

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, $a$ .

$a = 2$

$b = 4$

$k = 0$

$h = 0$

Step 4

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

$y = \pm \frac{1}{2} x + 0$

Step 5

Simplify $\frac{1}{2} x + 0$ .

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Step 5.1

Add $\frac{1}{2} x$ and $0$ .

$y = \frac{1}{2} x$

Step 5.2

Combine $\frac{1}{2}$ and $x$ .

$y = \frac{x}{2}$

Step 6

Simplify $- \frac{1}{2} x + 0$ .

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Step 6.1

Add $- \frac{1}{2} x$ and $0$ .

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

Step 6.2

Combine $x$ and $\frac{1}{2}$ .

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

Step 7

This hyperbola has two asymptotes.

$y = \frac{x}{2}, y = - \frac{x}{2}$

Step 8

The asymptotes are $y = \frac{x}{2}$ and $y = - \frac{x}{2}$ .

Asymptotes: $y = \frac{x}{2}, y = - \frac{x}{2}$

Step 9