Precalculus Examples

$\frac{y^{2}}{4} - \frac{x^{2}}{12} = 1$

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 $1$ .

$\frac{y^{2}}{4} - \frac{x^{2}}{12} = 1$

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, $a$ .

$a = 2$

$b = 2 \sqrt{3}$

$k = 0$

$h = 0$

Step 4

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

$(0, 0)$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

Simplify the expression.

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Step 5.3.1.1

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

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

Step 5.3.1.2

Apply the product rule to $2 \sqrt{3}$ .

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

Step 5.3.1.3

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

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

Step 5.3.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{4 + 4 {(3^{\frac{1}{2}})}^{2}}$

Step 5.3.2.2

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

$\sqrt{4 + 4 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 5.3.2.3

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

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

Step 5.3.2.4

Cancel the common factor of $2$ .

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Step 5.3.2.4.1

Cancel the common factor.

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

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{4 + 4 \cdot 3^{1}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{4 + 4 \cdot 3}$

Step 5.3.3

Simplify the expression.

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Step 5.3.3.1

Multiply $4$ by $3$ .

$\sqrt{4 + 12}$

Step 5.3.3.2

Add $4$ and $12$ .

$\sqrt{16}$

Step 5.3.3.3

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2}}$

Step 5.3.3.4

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

$4$

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.

$(0, 2)$

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.

$(0, - 2)$

Step 6.5

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

$(0, 2), (0, - 2)$

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 $k$ .

$(h, k + c)$

Step 7.2

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

$(0, 4)$

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.

$(0, - 4)$

Step 7.5

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

$(0, 4), (0, - 4)$

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.

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

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

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

Step 8.3.1.2

Apply the product rule to $2 \sqrt{3}$ .

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

Step 8.3.1.3

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

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

Step 8.3.1.4

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{4 + 4 {(3^{\frac{1}{2}})}^{2}}}{2}$

Step 8.3.1.4.2

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

$\frac{\sqrt{4 + 4 \cdot 3^{\frac{1}{2} \cdot 2}}}{2}$

Step 8.3.1.4.3

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

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

Step 8.3.1.4.4

Cancel the common factor of $2$ .

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Step 8.3.1.4.4.1

Cancel the common factor.

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

Step 8.3.1.4.4.2

Rewrite the expression.

$\frac{\sqrt{4 + 4 \cdot 3^{1}}}{2}$

Step 8.3.1.4.5

Evaluate the exponent.

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

Step 8.3.1.5

Multiply $4$ by $3$ .

$\frac{\sqrt{4 + 12}}{2}$

Step 8.3.1.6

Add $4$ and $12$ .

$\frac{\sqrt{16}}{2}$

Step 8.3.1.7

Rewrite $16$ as $4^{2}$ .

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

Step 8.3.1.8

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

$\frac{4}{2}$

Step 8.3.2

Divide $4$ by $2$ .

$2$

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.

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

Step 9.3

Simplify.

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Step 9.3.1

Cancel the common factor of $2$ and $4$ .

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Step 9.3.1.1

Factor $2$ out of $2 {\sqrt{3}}^{2}$ .

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

Step 9.3.1.2

Cancel the common factors.

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Step 9.3.1.2.1

Factor $2$ out of $4$ .

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

Step 9.3.1.2.2

Cancel the common factor.

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

Step 9.3.1.2.3

Rewrite the expression.

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

Step 9.3.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 9.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 9.3.2.2

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

$\frac{3^{\frac{1}{2} \cdot 2}}{2}$

Step 9.3.2.3

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

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

Step 9.3.2.4

Cancel the common factor of $2$ .

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Step 9.3.2.4.1

Cancel the common factor.

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

Step 9.3.2.4.2

Rewrite the expression.

$\frac{3^{1}}{2}$

Step 9.3.2.5

Evaluate the exponent.

$\frac{3}{2}$

Step 10

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

$y = \pm \frac{\sqrt{3}}{3} x + 0$

Step 11

Simplify $\frac{\sqrt{3}}{3} x + 0$ .

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Step 11.1

Add $\frac{\sqrt{3}}{3} x$ and $0$ .

$y = \frac{\sqrt{3}}{3} x$

Step 11.2

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

$y = \frac{\sqrt{3} x}{3}$

Step 12

Simplify $- \frac{\sqrt{3}}{3} x + 0$ .

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Step 12.1

Add $- \frac{\sqrt{3}}{3} x$ and $0$ .

$y = - \frac{\sqrt{3}}{3} x$

Step 12.2

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

$y = - \frac{x \sqrt{3}}{3}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{\sqrt{3} x}{3}, y = - \frac{x \sqrt{3}}{3}$

Step 14

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

Center: $(0, 0)$

Vertices: $(0, 2), (0, - 2)$

Foci: $(0, 4), (0, - 4)$

Eccentricity: $2$

Focal Parameter: $\frac{3}{2}$

Asymptotes: $y = \frac{\sqrt{3} x}{3}$ , $y = - \frac{x \sqrt{3}}{3}$

Step 15