Precalculus Examples

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

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 $1$ .

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

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, $a$ .

$a = 3$

$b = 1$

$k = 0$

$h = - 4$

Step 4

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

$(- 4, 0)$

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.

$\sqrt{{(3)}^{2} + {(1)}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

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

$\sqrt{9 + {(1)}^{2}}$

Step 5.3.2

One to any power is one.

$\sqrt{9 + 1}$

Step 5.3.3

Add $9$ and $1$ .

$\sqrt{10}$

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.

$(- 4, 3)$

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.

$(- 4, - 3)$

Step 6.5

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

$(- 4, 3), (- 4, - 3)$

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.

$(- 4, \sqrt{10})$

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.

$(- 4, - \sqrt{10})$

Step 7.5

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

$(- 4, \sqrt{10}), (- 4, - \sqrt{10})$

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.

$\frac{\sqrt{{(3)}^{2} + {(1)}^{2}}}{3}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

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

$\frac{\sqrt{9 + 1^{2}}}{3}$

Step 8.3.2

One to any power is one.

$\frac{\sqrt{9 + 1}}{3}$

Step 8.3.3

Add $9$ and $1$ .

$\frac{\sqrt{10}}{3}$

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.

$\frac{1^{2}}{\sqrt{10}}$

Step 9.3

Simplify.

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Step 9.3.1

One to any power is one.

$\frac{1}{\sqrt{10}}$

Step 9.3.2

Multiply $\frac{1}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{1}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

Multiply $\frac{1}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 9.3.3.2

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{\sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 9.3.3.3

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{\sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 9.3.3.4

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

$\frac{\sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 9.3.3.5

Add $1$ and $1$ .

$\frac{\sqrt{10}}{{\sqrt{10}}^{2}}$

Step 9.3.3.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 9.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{\sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

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

$\frac{\sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

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

$\frac{\sqrt{10}}{10^{\frac{2}{2}}}$

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.

$\frac{\sqrt{10}}{10^{\frac{2}{2}}}$

Step 9.3.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{10}}{10^{1}}$

Step 9.3.3.6.5

Evaluate the exponent.

$\frac{\sqrt{10}}{10}$

Step 10

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

$y = \pm 3 \cdot (x - (- 4)) + 0$

Step 11

Simplify to find the first asymptote.

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Step 11.1

Remove parentheses.

$y = 3 \cdot (x - (- 4)) + 0$

Step 11.2

Simplify $3 \cdot (x - (- 4)) + 0$ .

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Step 11.2.1

Simplify the expression.

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Step 11.2.1.1

Add $3 \cdot (x - (- 4))$ and $0$ .

$y = 3 \cdot (x - (- 4))$

Step 11.2.1.2

Multiply $- 1$ by $- 4$ .

$y = 3 \cdot (x + 4)$

Step 11.2.2

Apply the distributive property.

$y = 3 x + 3 \cdot 4$

Step 11.2.3

Multiply $3$ by $4$ .

$y = 3 x + 12$

Step 12

Simplify to find the second asymptote.

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Step 12.1

Remove parentheses.

$y = - 3 \cdot (x - (- 4)) + 0$

Step 12.2

Simplify $- 3 \cdot (x - (- 4)) + 0$ .

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Step 12.2.1

Simplify the expression.

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Step 12.2.1.1

Add $- 3 \cdot (x - (- 4))$ and $0$ .

$y = - 3 \cdot (x - (- 4))$

Step 12.2.1.2

Multiply $- 1$ by $- 4$ .

$y = - 3 \cdot (x + 4)$

Step 12.2.2

Apply the distributive property.

$y = - 3 x - 3 \cdot 4$

Step 12.2.3

Multiply $- 3$ by $4$ .

$y = - 3 x - 12$

Step 13

This hyperbola has two asymptotes.

$y = 3 x + 12, y = - 3 x - 12$

Step 14

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

Center: $(- 4, 0)$

Vertices: $(- 4, 3), (- 4, - 3)$

Foci: $(- 4, \sqrt{10}), (- 4, - \sqrt{10})$

Eccentricity: $\frac{\sqrt{10}}{3}$

Focal Parameter: $\frac{\sqrt{10}}{10}$

Asymptotes: $y = 3 x + 12$ , $y = - 3 x - 12$

Step 15