Precalculus Examples

$\frac{{(x - 2)}^{2}}{36} - \frac{{(y - 3)}^{2}}{9} = 1$

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 $1$ .

$\frac{{(x - 2)}^{2}}{36} - \frac{{(y - 3)}^{2}}{9} = 1$

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, $a$ .

$a = 6$

$b = 3$

$k = 3$

$h = 2$

Step 4

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

$y = \pm \frac{1}{2} \cdot (x - (2)) + 3$

Step 5

Simplify to find the first asymptote.

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Step 5.1

Remove parentheses.

$y = \frac{1}{2} \cdot (x - (2)) + 3$

Step 5.2

Simplify $\frac{1}{2} \cdot (x - (2)) + 3$ .

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $- 1$ by $2$ .

$y = \frac{1}{2} \cdot (x - 2) + 3$

Step 5.2.1.2

Apply the distributive property.

$y = \frac{1}{2} x + \frac{1}{2} \cdot - 2 + 3$

Step 5.2.1.3

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

$y = \frac{x}{2} + \frac{1}{2} \cdot - 2 + 3$

Step 5.2.1.4

Cancel the common factor of $2$ .

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Step 5.2.1.4.1

Factor $2$ out of $- 2$ .

$y = \frac{x}{2} + \frac{1}{2} \cdot (2 (- 1)) + 3$

Step 5.2.1.4.2

Cancel the common factor.

$y = \frac{x}{2} + \frac{1}{2} \cdot (2 \cdot - 1) + 3$

Step 5.2.1.4.3

Rewrite the expression.

$y = \frac{x}{2} - 1 + 3$

Step 5.2.2

Add $- 1$ and $3$ .

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

Step 6

Simplify to find the second asymptote.

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Step 6.1

Remove parentheses.

$y = - \frac{1}{2} \cdot (x - (2)) + 3$

Step 6.2

Simplify $- \frac{1}{2} \cdot (x - (2)) + 3$ .

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $- 1$ by $2$ .

$y = - \frac{1}{2} \cdot (x - 2) + 3$

Step 6.2.1.2

Apply the distributive property.

$y = - \frac{1}{2} x - \frac{1}{2} \cdot - 2 + 3$

Step 6.2.1.3

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

$y = - \frac{x}{2} - \frac{1}{2} \cdot - 2 + 3$

Step 6.2.1.4

Cancel the common factor of $2$ .

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Step 6.2.1.4.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y = - \frac{x}{2} + \frac{- 1}{2} \cdot - 2 + 3$

Step 6.2.1.4.2

Factor $2$ out of $- 2$ .

$y = - \frac{x}{2} + \frac{- 1}{2} \cdot (2 (- 1)) + 3$

Step 6.2.1.4.3

Cancel the common factor.

$y = - \frac{x}{2} + \frac{- 1}{2} \cdot (2 \cdot - 1) + 3$

Step 6.2.1.4.4

Rewrite the expression.

$y = - \frac{x}{2} - 1 \cdot - 1 + 3$

Step 6.2.1.5

Multiply $- 1$ by $- 1$ .

$y = - \frac{x}{2} + 1 + 3$

Step 6.2.2

Add $1$ and $3$ .

$y = - \frac{x}{2} + 4$

Step 7

This hyperbola has two asymptotes.

$y = \frac{x}{2} + 2, y = - \frac{x}{2} + 4$

Step 8

The asymptotes are $y = \frac{x}{2} + 2$ and $y = - \frac{x}{2} + 4$ .

Asymptotes: $y = \frac{x}{2} + 2, y = - \frac{x}{2} + 4$

Step 9