Algebra Examples

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

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

Step 1

Simplify each term in the equation in order to set the right side equal to

1

$1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be

1

$1$ .

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

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

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

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

Step 3

Match the values in this hyperbola to those of the standard form. The variable

h

$h$ represents the x-offset from the origin,

k

$k$ represents the y-offset from origin,

a

$a$ .

a = 2

$a = 2$

b = 1

$b = 1$

k = 0

$k = 0$

h = 0

$h = 0$

Step 4

The center of a hyperbola follows the form of

(h, k)

$(h, k)$ . Substitute in the values of

h

$h$ and

k

$k$ .

(0, 0)

$(0, 0)$

Step 5

Find

c

$c$ , 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}}

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

Step 5.2

Substitute the values of

a

$a$ and

b

$b$ in the formula.

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

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

Step 5.3

Simplify.

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Step 5.3.1

Raise

2

$2$ to the power of

2

$2$ .

\sqrt{4 + {(1)}^{2}}

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

Step 5.3.2

One to any power is one.

\sqrt{4 + 1}

$\sqrt{4 + 1}$

Step 5.3.3

Add

4

$4$ and

1

$1$ .

\sqrt{5}

$\sqrt{5}$

\sqrt{5}

$\sqrt{5}$

\sqrt{5}

$\sqrt{5}$

Step 6

Find the vertices.

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Step 6.1

The first vertex of a hyperbola can be found by adding

a

$a$ to

h

$h$ .

(h + a, k)

$(h + a, k)$

Step 6.2

Substitute the known values of

h

$h$ ,

a

$a$ , and

$k$ into the formula and simplify.

$(2, 0)$

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.

$(- 2, 0)$

Step 6.5

The vertices of a hyperbola follow the form of

$(h \pm a, k)$ . Hyperbolas have two vertices.

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

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.

$(\sqrt{5}, 0)$

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.

$(- \sqrt{5}, 0)$

Step 7.5

The foci of a hyperbola follow the form of

$(h \pm \sqrt{a^{2} + b^{2}}, k)$ . Hyperbolas have two foci.

$(\sqrt{5}, 0), (- \sqrt{5}, 0)$

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.

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

Step 8.3

Simplify the numerator.

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Step 8.3.1

Raise

$2$ to the power of

$2$ .

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

Step 8.3.2

One to any power is one.

$\frac{\sqrt{4 + 1}}{2}$

Step 8.3.3

Add

$4$ and

$1$ .

$\frac{\sqrt{5}}{2}$

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.

$\frac{1^{2}}{\sqrt{5}}$

Step 9.3

Simplify.

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Step 9.3.1

One to any power is one.

$\frac{1}{\sqrt{5}}$

Step 9.3.2

Multiply

$\frac{1}{\sqrt{5}}$ by

$\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

Multiply

$\frac{1}{\sqrt{5}}$ by

$\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 9.3.3.2

Raise

$\sqrt{5}$ to the power of

$1$ .

$\frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 9.3.3.3

Raise

$\sqrt{5}$ to the power of

$1$ .

$\frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 9.3.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 9.3.3.5

Add

$1$ and

$1$ .

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

Step 9.3.3.6

Rewrite

${\sqrt{5}}^{2}$ as

$5$ .

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Step 9.3.3.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{5}$ as

$5^{\frac{1}{2}}$ .

$\frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

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.

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

Step 9.3.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{5}}{5^{1}}$

Step 9.3.3.6.5

Evaluate the exponent.

$\frac{\sqrt{5}}{5}$

Step 10

The asymptotes follow the form

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

$y = \pm \frac{1}{2} x + 0$

Step 11

Simplify

$\frac{1}{2} x + 0$ .

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Step 11.1

Add

$\frac{1}{2} x$ and

$0$ .

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

Step 11.2

Combine

$\frac{1}{2}$ and

$x$ .

$y = \frac{x}{2}$

Step 12

Simplify

$- \frac{1}{2} x + 0$ .

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Step 12.1

Add

$- \frac{1}{2} x$ and

$0$ .

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

Step 12.2

Combine

$x$ and

$\frac{1}{2}$ .

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

Step 13

This hyperbola has two asymptotes.

$y = \frac{x}{2}, y = - \frac{x}{2}$

Step 14

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

Center:

$(0, 0)$

Vertices:

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

Foci:

$(\sqrt{5}, 0), (- \sqrt{5}, 0)$

Eccentricity:

$\frac{\sqrt{5}}{2}$

Focal Parameter:

$\frac{\sqrt{5}}{5}$

Asymptotes:

$y = \frac{x}{2}$ ,

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

Step 15