Trigonometry Examples

\frac{y^{2}}{9} - \frac{x^{2}}{36} = 1

$\frac{y^{2}}{9} - \frac{x^{2}}{36} = 1$

Step 1

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}}{9} - \frac{x^{2}}{36} = 1

$\frac{y^{2}}{9} - \frac{x^{2}}{36} = 1$

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 = 3

$a = 3$

b = 6

$b = 6$

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 \frac{1}{2} x + 0

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

Step 5

Simplify

\frac{1}{2} x + 0

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

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Step 5.1

Add

\frac{1}{2} x

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

0

$0$ .

y = \frac{1}{2} x

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

Step 5.2

Combine

\frac{1}{2}

$\frac{1}{2}$ and

x

$x$ .

y = \frac{x}{2}

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

y = \frac{x}{2}

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

Step 6

Simplify

- \frac{1}{2} x + 0

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

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Step 6.1

Add

- \frac{1}{2} x

$- \frac{1}{2} x$ and

0

$0$ .

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

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

Step 6.2

Combine

x

$x$ and

\frac{1}{2}

$\frac{1}{2}$ .

y = - \frac{x}{2}

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

y = - \frac{x}{2}

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

Step 7

This hyperbola has two asymptotes.

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

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

Step 8

The asymptotes are

y = \frac{x}{2}

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

y = - \frac{x}{2}

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

Asymptotes:

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

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

Step 9