Precalculus Examples

$\frac{{(x + 3)}^{2}}{144} - \frac{{(y - 2)}^{2}}{25} = 1$

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 $1$ .

$\frac{{(x + 3)}^{2}}{144} - \frac{{(y - 2)}^{2}}{25} = 1$

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.

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

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, $a$ .

$a = 12$

$b = 5$

$k = 2$

$h = - 3$

Step 4

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

$(- 3, 2)$

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.

$\sqrt{{(12)}^{2} + {(5)}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Raise $12$ to the power of $2$ .

$\sqrt{144 + {(5)}^{2}}$

Step 5.3.2

Raise $5$ to the power of $2$ .

$\sqrt{144 + 25}$

Step 5.3.3

Add $144$ and $25$ .

$\sqrt{169}$

Step 5.3.4

Rewrite $169$ as $13^{2}$ .

$\sqrt{13^{2}}$

Step 5.3.5

Pull terms out from under the radical, assuming positive real numbers.

$13$

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 $h$ .

$(h + a, k)$

Step 6.2

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

$(9, 2)$

Step 6.3

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

$(h - a, k)$

Step 6.4

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

$(- 15, 2)$

Step 6.5

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

$(9, 2), (- 15, 2)$

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 $h$ .

$(h + c, k)$

Step 7.2

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

$(10, 2)$

Step 7.3

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

$(h - c, k)$

Step 7.4

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

$(- 16, 2)$

Step 7.5

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

$(10, 2), (- 16, 2)$

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.

$\frac{\sqrt{{(12)}^{2} + {(5)}^{2}}}{12}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Raise $12$ to the power of $2$ .

$\frac{\sqrt{144 + 5^{2}}}{12}$

Step 8.3.2

Raise $5$ to the power of $2$ .

$\frac{\sqrt{144 + 25}}{12}$

Step 8.3.3

Add $144$ and $25$ .

$\frac{\sqrt{169}}{12}$

Step 8.3.4

Rewrite $169$ as $13^{2}$ .

$\frac{\sqrt{13^{2}}}{12}$

Step 8.3.5

Pull terms out from under the radical, assuming positive real numbers.

$\frac{13}{12}$

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.

$\frac{5^{2}}{13}$

Step 9.3

Raise $5$ to the power of $2$ .

$\frac{25}{13}$

Step 10

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

$y = \pm \frac{5}{12} \cdot (x - (- 3)) + 2$

Step 11

Simplify to find the first asymptote.

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Step 11.1

Remove parentheses.

$y = \frac{5}{12} \cdot (x - (- 3)) + 2$

Step 11.2

Simplify $\frac{5}{12} \cdot (x - (- 3)) + 2$ .

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Multiply $- 1$ by $- 3$ .

$y = \frac{5}{12} \cdot (x + 3) + 2$

Step 11.2.1.2

Apply the distributive property.

$y = \frac{5}{12} x + \frac{5}{12} \cdot 3 + 2$

Step 11.2.1.3

Combine $\frac{5}{12}$ and $x$ .

$y = \frac{5 x}{12} + \frac{5}{12} \cdot 3 + 2$

Step 11.2.1.4

Cancel the common factor of $3$ .

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Step 11.2.1.4.1

Factor $3$ out of $12$ .

$y = \frac{5 x}{12} + \frac{5}{3 (4)} \cdot 3 + 2$

Step 11.2.1.4.2

Cancel the common factor.

$y = \frac{5 x}{12} + \frac{5}{3 \cdot 4} \cdot 3 + 2$

Step 11.2.1.4.3

Rewrite the expression.

$y = \frac{5 x}{12} + \frac{5}{4} + 2$

Step 11.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{5 x}{12} + \frac{5}{4} + 2 \cdot \frac{4}{4}$

Step 11.2.3

Combine $2$ and $\frac{4}{4}$ .

$y = \frac{5 x}{12} + \frac{5}{4} + \frac{2 \cdot 4}{4}$

Step 11.2.4

Combine the numerators over the common denominator.

$y = \frac{5 x}{12} + \frac{5 + 2 \cdot 4}{4}$

Step 11.2.5

Simplify the numerator.

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Step 11.2.5.1

Multiply $2$ by $4$ .

$y = \frac{5 x}{12} + \frac{5 + 8}{4}$

Step 11.2.5.2

Add $5$ and $8$ .

$y = \frac{5 x}{12} + \frac{13}{4}$

Step 12

Simplify to find the second asymptote.

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Step 12.1

Remove parentheses.

$y = - \frac{5}{12} \cdot (x - (- 3)) + 2$

Step 12.2

Simplify $- \frac{5}{12} \cdot (x - (- 3)) + 2$ .

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Multiply $- 1$ by $- 3$ .

$y = - \frac{5}{12} \cdot (x + 3) + 2$

Step 12.2.1.2

Apply the distributive property.

$y = - \frac{5}{12} x - \frac{5}{12} \cdot 3 + 2$

Step 12.2.1.3

Combine $x$ and $\frac{5}{12}$ .

$y = - \frac{x \cdot 5}{12} - \frac{5}{12} \cdot 3 + 2$

Step 12.2.1.4

Cancel the common factor of $3$ .

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Step 12.2.1.4.1

Move the leading negative in $- \frac{5}{12}$ into the numerator.

$y = - \frac{x \cdot 5}{12} + \frac{- 5}{12} \cdot 3 + 2$

Step 12.2.1.4.2

Factor $3$ out of $12$ .

$y = - \frac{x \cdot 5}{12} + \frac{- 5}{3 (4)} \cdot 3 + 2$

Step 12.2.1.4.3

Cancel the common factor.

$y = - \frac{x \cdot 5}{12} + \frac{- 5}{3 \cdot 4} \cdot 3 + 2$

Step 12.2.1.4.4

Rewrite the expression.

$y = - \frac{x \cdot 5}{12} + \frac{- 5}{4} + 2$

Step 12.2.1.5

Simplify each term.

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Step 12.2.1.5.1

Move $5$ to the left of $x$ .

$y = - \frac{5 \cdot x}{12} + \frac{- 5}{4} + 2$

Step 12.2.1.5.2

Move the negative in front of the fraction.

$y = - \frac{5 x}{12} - \frac{5}{4} + 2$

Step 12.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = - \frac{5 x}{12} - \frac{5}{4} + 2 \cdot \frac{4}{4}$

Step 12.2.3

Combine $2$ and $\frac{4}{4}$ .

$y = - \frac{5 x}{12} - \frac{5}{4} + \frac{2 \cdot 4}{4}$

Step 12.2.4

Combine the numerators over the common denominator.

$y = - \frac{5 x}{12} + \frac{- 5 + 2 \cdot 4}{4}$

Step 12.2.5

Simplify the numerator.

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Step 12.2.5.1

Multiply $2$ by $4$ .

$y = - \frac{5 x}{12} + \frac{- 5 + 8}{4}$

Step 12.2.5.2

Add $- 5$ and $8$ .

$y = - \frac{5 x}{12} + \frac{3}{4}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{5 x}{12} + \frac{13}{4}, y = - \frac{5 x}{12} + \frac{3}{4}$

Step 14

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

Center: $(- 3, 2)$

Vertices: $(9, 2), (- 15, 2)$

Foci: $(10, 2), (- 16, 2)$

Eccentricity: $\frac{13}{12}$

Focal Parameter: $\frac{25}{13}$

Asymptotes: $y = \frac{5 x}{12} + \frac{13}{4}$ , $y = - \frac{5 x}{12} + \frac{3}{4}$

Step 15