Pre-Algebra Examples

$\frac{{(x - 9)}^{2}}{9} - \frac{{(y - 10)}^{2}}{7} = 1$

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 $1$ .

$\frac{{(x - 9)}^{2}}{9} - \frac{{(y - 10)}^{2}}{7} = 1$

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, $a$ .

$a = 3$

$b = \sqrt{7}$

$k = 10$

$h = 9$

Step 4

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

$(9, 10)$

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.

$\sqrt{{(3)}^{2} + {(\sqrt{7})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

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

$\sqrt{9 + {(\sqrt{7})}^{2}}$

Step 5.3.2

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{9 + {(7^{\frac{1}{2}})}^{2}}$

Step 5.3.2.2

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

$\sqrt{9 + 7^{\frac{1}{2} \cdot 2}}$

Step 5.3.2.3

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

$\sqrt{9 + 7^{\frac{2}{2}}}$

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.

$\sqrt{9 + 7^{\frac{2}{2}}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{9 + 7^{1}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{9 + 7}$

Step 5.3.3

Simplify the expression.

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Step 5.3.3.1

Add $9$ and $7$ .

$\sqrt{16}$

Step 5.3.3.2

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

$\sqrt{4^{2}}$

Step 5.3.3.3

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 $h$ .

$(h + a, k)$

Step 6.2

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

$(12, 10)$

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.

$(6, 10)$

Step 6.5

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

$(12, 10), (6, 10)$

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.

$(13, 10)$

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.

$(5, 10)$

Step 7.5

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

$(13, 10), (5, 10)$

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.

$\frac{\sqrt{{(3)}^{2} + {(\sqrt{7})}^{2}}}{3}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

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

$\frac{\sqrt{9 + {\sqrt{7}}^{2}}}{3}$

Step 8.3.2

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 8.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\sqrt{9 + {(7^{\frac{1}{2}})}^{2}}}{3}$

Step 8.3.2.2

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

$\frac{\sqrt{9 + 7^{\frac{1}{2} \cdot 2}}}{3}$

Step 8.3.2.3

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

$\frac{\sqrt{9 + 7^{\frac{2}{2}}}}{3}$

Step 8.3.2.4

Cancel the common factor of $2$ .

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Step 8.3.2.4.1

Cancel the common factor.

$\frac{\sqrt{9 + 7^{\frac{2}{2}}}}{3}$

Step 8.3.2.4.2

Rewrite the expression.

$\frac{\sqrt{9 + 7^{1}}}{3}$

Step 8.3.2.5

Evaluate the exponent.

$\frac{\sqrt{9 + 7}}{3}$

Step 8.3.3

Add $9$ and $7$ .

$\frac{\sqrt{16}}{3}$

Step 8.3.4

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

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

Step 8.3.5

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

$\frac{4}{3}$

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.

$\frac{{\sqrt{7}}^{2}}{4}$

Step 9.3

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 9.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{{(7^{\frac{1}{2}})}^{2}}{4}$

Step 9.3.2

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

$\frac{7^{\frac{1}{2} \cdot 2}}{4}$

Step 9.3.3

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

$\frac{7^{\frac{2}{2}}}{4}$

Step 9.3.4

Cancel the common factor of $2$ .

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Step 9.3.4.1

Cancel the common factor.

$\frac{7^{\frac{2}{2}}}{4}$

Step 9.3.4.2

Rewrite the expression.

$\frac{7^{1}}{4}$

Step 9.3.5

Evaluate the exponent.

$\frac{7}{4}$

Step 10

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

$y = \pm \frac{\sqrt{7}}{3} \cdot (x - (9)) + 10$

Step 11

Simplify to find the first asymptote.

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Step 11.1

Remove parentheses.

$y = \frac{\sqrt{7}}{3} \cdot (x - (9)) + 10$

Step 11.2

Simplify each term.

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Step 11.2.1

Multiply $- 1$ by $9$ .

$y = \frac{\sqrt{7}}{3} \cdot (x - 9) + 10$

Step 11.2.2

Apply the distributive property.

$y = \frac{\sqrt{7}}{3} x + \frac{\sqrt{7}}{3} \cdot - 9 + 10$

Step 11.2.3

Combine $\frac{\sqrt{7}}{3}$ and $x$ .

$y = \frac{\sqrt{7} x}{3} + \frac{\sqrt{7}}{3} \cdot - 9 + 10$

Step 11.2.4

Cancel the common factor of $3$ .

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Step 11.2.4.1

Factor $3$ out of $- 9$ .

$y = \frac{\sqrt{7} x}{3} + \frac{\sqrt{7}}{3} \cdot (3 (- 3)) + 10$

Step 11.2.4.2

Cancel the common factor.

$y = \frac{\sqrt{7} x}{3} + \frac{\sqrt{7}}{3} \cdot (3 \cdot - 3) + 10$

Step 11.2.4.3

Rewrite the expression.

$y = \frac{\sqrt{7} x}{3} + \sqrt{7} \cdot - 3 + 10$

Step 11.2.5

Move $- 3$ to the left of $\sqrt{7}$ .

$y = \frac{\sqrt{7} x}{3} - 3 \sqrt{7} + 10$

Step 12

Simplify to find the second asymptote.

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Step 12.1

Remove parentheses.

$y = - \frac{\sqrt{7}}{3} \cdot (x - (9)) + 10$

Step 12.2

Simplify each term.

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Step 12.2.1

Multiply $- 1$ by $9$ .

$y = - \frac{\sqrt{7}}{3} \cdot (x - 9) + 10$

Step 12.2.2

Apply the distributive property.

$y = - \frac{\sqrt{7}}{3} x - \frac{\sqrt{7}}{3} \cdot - 9 + 10$

Step 12.2.3

Combine $x$ and $\frac{\sqrt{7}}{3}$ .

$y = - \frac{x \sqrt{7}}{3} - \frac{\sqrt{7}}{3} \cdot - 9 + 10$

Step 12.2.4

Cancel the common factor of $3$ .

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Step 12.2.4.1

Move the leading negative in $- \frac{\sqrt{7}}{3}$ into the numerator.

$y = - \frac{x \sqrt{7}}{3} + \frac{- \sqrt{7}}{3} \cdot - 9 + 10$

Step 12.2.4.2

Factor $3$ out of $- 9$ .

$y = - \frac{x \sqrt{7}}{3} + \frac{- \sqrt{7}}{3} \cdot (3 (- 3)) + 10$

Step 12.2.4.3

Cancel the common factor.

$y = - \frac{x \sqrt{7}}{3} + \frac{- \sqrt{7}}{3} \cdot (3 \cdot - 3) + 10$

Step 12.2.4.4

Rewrite the expression.

$y = - \frac{x \sqrt{7}}{3} - \sqrt{7} \cdot - 3 + 10$

Step 12.2.5

Multiply $- 3$ by $- 1$ .

$y = - \frac{x \sqrt{7}}{3} + 3 \sqrt{7} + 10$

Step 13

This hyperbola has two asymptotes.

$y = \frac{\sqrt{7} x}{3} - 3 \sqrt{7} + 10, y = - \frac{x \sqrt{7}}{3} + 3 \sqrt{7} + 10$

Step 14

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

Center: $(9, 10)$

Vertices: $(12, 10), (6, 10)$

Foci: $(13, 10), (5, 10)$

Eccentricity: $\frac{4}{3}$

Focal Parameter: $\frac{7}{4}$

Asymptotes: $y = \frac{\sqrt{7} x}{3} - 3 \sqrt{7} + 10$ , $y = - \frac{x \sqrt{7}}{3} + 3 \sqrt{7} + 10$

Step 15