Precalculus Examples

$\frac{{(y - 1)}^{2}}{4} - \frac{{(x + 2)}^{2}}{9} = 1$

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 $1$ .

$\frac{{(y - 1)}^{2}}{4} - \frac{{(x + 2)}^{2}}{9} = 1$

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find vertices and 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, $a$ .

$a = 2$

$b = 3$

$k = 1$

$h = - 2$

Step 4

The center of a hyperbola follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(- 2, 1)$

Step 5

Find $c$ , the distance from the center to a focus.

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Step 5.1

Find the distance from the center to a focus of the hyperbola by using the following formula.

$\sqrt{a^{2} + b^{2}}$

Step 5.2

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(2)}^{2} + {(3)}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Raise $2$ to the power of $2$ .

$\sqrt{4 + {(3)}^{2}}$

Step 5.3.2

Raise $3$ to the power of $2$ .

$\sqrt{4 + 9}$

Step 5.3.3

Add $4$ and $9$ .

$\sqrt{13}$

Step 6

Find the vertices.

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Step 6.1

The first vertex of a hyperbola can be found by adding $a$ to $k$ .

$(h, k + a)$

Step 6.2

Substitute the known values of $h$ , $a$ , and $k$ into the formula and simplify.

$(- 2, 3)$

Step 6.3

The second vertex of a hyperbola can be found by subtracting $a$ from $k$ .

$(h, k - a)$

Step 6.4

Substitute the known values of $h$ , $a$ , and $k$ into the formula and simplify.

$(- 2, - 1)$

Step 6.5

The vertices of a hyperbola follow the form of $(h, k \pm a)$ . Hyperbolas have two vertices.

$(- 2, 3), (- 2, - 1)$

Step 7

Find the foci.

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Step 7.1

The first focus of a hyperbola can be found by adding $c$ to $k$ .

$(h, k + c)$

Step 7.2

Substitute the known values of $h$ , $c$ , and $k$ into the formula and simplify.

$(- 2, 1 + \sqrt{13})$

Step 7.3

The second focus of a hyperbola can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Step 7.4

Substitute the known values of $h$ , $c$ , and $k$ into the formula and simplify.

$(- 2, 1 - \sqrt{13})$

Step 7.5

The foci of a hyperbola follow the form of $(h, k \pm \sqrt{a^{2} + b^{2}})$ . Hyperbolas have two foci.

$(- 2, 1 + \sqrt{13}), (- 2, 1 - \sqrt{13})$

Step 8

Find the eccentricity.

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Step 8.1

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} + b^{2}}}{a}$

Step 8.2

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(2)}^{2} + {(3)}^{2}}}{2}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Raise $2$ to the power of $2$ .

$\frac{\sqrt{4 + 3^{2}}}{2}$

Step 8.3.2

Raise $3$ to the power of $2$ .

$\frac{\sqrt{4 + 9}}{2}$

Step 8.3.3

Add $4$ and $9$ .

$\frac{\sqrt{13}}{2}$

Step 9

Find the focal parameter.

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Step 9.1

Find the value of the focal parameter of the hyperbola by using the following formula.

$\frac{b^{2}}{\sqrt{a^{2} + b^{2}}}$

Step 9.2

Substitute the values of $b$ and $\sqrt{a^{2} + b^{2}}$ in the formula.

$\frac{3^{2}}{\sqrt{13}}$

Step 9.3

Simplify.

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Step 9.3.1

Raise $3$ to the power of $2$ .

$\frac{9}{\sqrt{13}}$

Step 9.3.2

Multiply $\frac{9}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{9}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}$

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

Multiply $\frac{9}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{9 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 9.3.3.2

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{9 \sqrt{13}}{{\sqrt{13}}^{1} \sqrt{13}}$

Step 9.3.3.3

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{9 \sqrt{13}}{{\sqrt{13}}^{1} {\sqrt{13}}^{1}}$

Step 9.3.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

Step 9.3.3.5

Add $1$ and $1$ .

$\frac{9 \sqrt{13}}{{\sqrt{13}}^{2}}$

Step 9.3.3.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 9.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{9 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{9 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

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

$\frac{9 \sqrt{13}}{13^{\frac{2}{2}}}$

Step 9.3.3.6.4

Cancel the common factor of $2$ .

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Step 9.3.3.6.4.1

Cancel the common factor.

$\frac{9 \sqrt{13}}{13^{\frac{2}{2}}}$

Step 9.3.3.6.4.2

Rewrite the expression.

$\frac{9 \sqrt{13}}{13^{1}}$

Step 9.3.3.6.5

Evaluate the exponent.

$\frac{9 \sqrt{13}}{13}$

Step 10

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

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

Step 11

Simplify to find the first asymptote.

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Step 11.1

Remove parentheses.

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

Step 11.2

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

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Multiply $- 1$ by $- 2$ .

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

Step 11.2.1.2

Apply the distributive property.

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

Step 11.2.1.3

Combine $\frac{2}{3}$ and $x$ .

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

Step 11.2.1.4

Multiply $\frac{2}{3} \cdot 2$ .

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Step 11.2.1.4.1

Combine $\frac{2}{3}$ and $2$ .

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

Step 11.2.1.4.2

Multiply $2$ by $2$ .

$y = \frac{2 x}{3} + \frac{4}{3} + 1$

Step 11.2.2

Simplify the expression.

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Step 11.2.2.1

Write $1$ as a fraction with a common denominator.

$y = \frac{2 x}{3} + \frac{4}{3} + \frac{3}{3}$

Step 11.2.2.2

Combine the numerators over the common denominator.

$y = \frac{2 x}{3} + \frac{4 + 3}{3}$

Step 11.2.2.3

Add $4$ and $3$ .

$y = \frac{2 x}{3} + \frac{7}{3}$

Step 12

Simplify to find the second asymptote.

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Step 12.1

Remove parentheses.

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

Step 12.2

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

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Multiply $- 1$ by $- 2$ .

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

Step 12.2.1.2

Apply the distributive property.

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

Step 12.2.1.3

Combine $x$ and $\frac{2}{3}$ .

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

Step 12.2.1.4

Multiply $- \frac{2}{3} \cdot 2$ .

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Step 12.2.1.4.1

Multiply $2$ by $- 1$ .

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

Step 12.2.1.4.2

Combine $- 2$ and $\frac{2}{3}$ .

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

Step 12.2.1.4.3

Multiply $- 2$ by $2$ .

$y = - \frac{x \cdot 2}{3} + \frac{- 4}{3} + 1$

Step 12.2.1.5

Simplify each term.

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Step 12.2.1.5.1

Move $2$ to the left of $x$ .

$y = - \frac{2 \cdot x}{3} + \frac{- 4}{3} + 1$

Step 12.2.1.5.2

Move the negative in front of the fraction.

$y = - \frac{2 x}{3} - \frac{4}{3} + 1$

Step 12.2.2

Simplify the expression.

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Step 12.2.2.1

Write $1$ as a fraction with a common denominator.

$y = - \frac{2 x}{3} - \frac{4}{3} + \frac{3}{3}$

Step 12.2.2.2

Combine the numerators over the common denominator.

$y = - \frac{2 x}{3} + \frac{- 4 + 3}{3}$

Step 12.2.2.3

Add $- 4$ and $3$ .

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

Step 12.2.2.4

Move the negative in front of the fraction.

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

Step 13

This hyperbola has two asymptotes.

$y = \frac{2 x}{3} + \frac{7}{3}, y = - \frac{2 x}{3} - \frac{1}{3}$

Step 14

These values represent the important values for graphing and analyzing a hyperbola.

Center: $(- 2, 1)$

Vertices: $(- 2, 3), (- 2, - 1)$

Foci: $(- 2, 1 + \sqrt{13}), (- 2, 1 - \sqrt{13})$

Eccentricity: $\frac{\sqrt{13}}{2}$

Focal Parameter: $\frac{9 \sqrt{13}}{13}$

Asymptotes: $y = \frac{2 x}{3} + \frac{7}{3}$ , $y = - \frac{2 x}{3} - \frac{1}{3}$

Step 15