Precalculus Examples

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

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

Step 1

Find the standard form of the hyperbola.

Tap for more steps...

Step 1.1

Divide each term by

16

$16$ to make the right side equal to one.

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

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

Step 1.2

Simplify each term in the equation in order to set the right side equal to

1

$1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be

1

$1$ .

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

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

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

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

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

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

Step 3

Match the values in this hyperbola to those of the standard form. The variable

h

$h$ represents the x-offset from the origin,

k

$k$ represents the y-offset from origin,

a

$a$ .

a = 4

$a = 4$

b = 4

$b = 4$

k = 0

$k = 0$

h = 0

$h = 0$

Step 4

The asymptotes follow the form

y = \pm \frac{a (x - h)}{b} + k

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

y = \pm 1 \cdot x + 0

$y = \pm 1 \cdot x + 0$

Step 5

Simplify

1 \cdot x + 0

$1 \cdot x + 0$ .

Tap for more steps...

Step 5.1

Add

1 \cdot x

$1 \cdot x$ and

0

$0$ .

y = 1 \cdot x

$y = 1 \cdot x$

Step 5.2

Multiply

x

$x$ by

1

$1$ .

y = x

$y = x$

y = x

$y = x$

Step 6

Simplify

- 1 \cdot x + 0

$- 1 \cdot x + 0$ .

Tap for more steps...

Step 6.1

Add

- 1 \cdot x

$- 1 \cdot x$ and

0

$0$ .

y = - 1 \cdot x

$y = - 1 \cdot x$

Step 6.2

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

y = - x

$y = - x$

y = - x

$y = - x$

Step 7

This hyperbola has two asymptotes.

y = x, y = - x

$y = x, y = - x$

Step 8

The asymptotes are

y = x

$y = x$ and

y = - x

$y = - x$ .

Asymptotes:

y = x, y = - x

$y = x, y = - x$

Step 9