Precalculus Examples

y = h (x)

$y = h (x)$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Move all terms containing variables to the left side of the equation.

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Step 1.1.1

Subtract

h (x)

$h (x)$ from both sides of the equation.

y - h x = 0

$y - h x = 0$

Step 1.1.2

Reorder

y

$y$ and

- h x

$- h x$ .

- h x + y = 0

$- h x + y = 0$

- h x + y = 0

$- h x + y = 0$

Step 1.2

Divide each term by

0

$0$ to make the right side equal to one.

- \frac{h x}{0} + \frac{y}{0} = \frac{0}{0}

$- \frac{h x}{0} + \frac{y}{0} = \frac{0}{0}$

Step 1.3

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

$1$ .

y - h x = 1

$y - h x = 1$

y - h x = 1

$y - h x = 1$

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 = 1

$a = 1$

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

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

Step 5.3

Simplify.

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Step 5.3.1

One to any power is one.

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

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

Step 5.3.2

One to any power is one.

\sqrt{1 + 1}

$\sqrt{1 + 1}$

Step 5.3.3

Add

1

$1$ and

1

$1$ .

\sqrt{2}

$\sqrt{2}$

\sqrt{2}

$\sqrt{2}$

\sqrt{2}

$\sqrt{2}$

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

$k$ into the formula and simplify.

(1, 0)

$(1, 0)$

Step 6.3

The second vertex of a hyperbola can be found by subtracting

a

$a$ from

h

$h$ .

(h - a, k)

$(h - a, k)$

Step 6.4

Substitute the known values of

h

$h$ ,

a

$a$ , and

k

$k$ into the formula and simplify.

(- 1, 0)

$(- 1, 0)$

Step 6.5

The vertices of a hyperbola follow the form of

(h \pm a, k)

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

(1, 0), (- 1, 0)

$(1, 0), (- 1, 0)$

(1, 0), (- 1, 0)

$(1, 0), (- 1, 0)$

Step 7

Find the foci.

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Step 7.1

The first focus of a hyperbola can be found by adding

c

$c$ to

h

$h$ .

(h + c, k)

$(h + c, k)$

Step 7.2

Substitute the known values of

h

$h$ ,

c

$c$ , and

k

$k$ into the formula and simplify.

(\sqrt{2}, 0)

$(\sqrt{2}, 0)$

Step 7.3

The second focus of a hyperbola can be found by subtracting

c

$c$ from

h

$h$ .

(h - c, k)

$(h - c, k)$

Step 7.4

Substitute the known values of

h

$h$ ,

c

$c$ , and

k

$k$ into the formula and simplify.

(- \sqrt{2}, 0)

$(- \sqrt{2}, 0)$

Step 7.5

The foci of a hyperbola follow the form of

(h \pm \sqrt{a^{2} + b^{2}}, k)

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

(\sqrt{2}, 0), (- \sqrt{2}, 0)

$(\sqrt{2}, 0), (- \sqrt{2}, 0)$

(\sqrt{2}, 0), (- \sqrt{2}, 0)

$(\sqrt{2}, 0), (- \sqrt{2}, 0)$

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}

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

Step 8.2

Substitute the values of

a

$a$ and

b

$b$ into the formula.

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

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

Step 8.3

Simplify.

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Step 8.3.1

Divide

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

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

1

$1$ .

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

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

Step 8.3.2

One to any power is one.

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

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

Step 8.3.3

One to any power is one.

\sqrt{1 + 1}

$\sqrt{1 + 1}$

Step 8.3.4

Add

1

$1$ and

1

$1$ .

\sqrt{2}

$\sqrt{2}$

\sqrt{2}

$\sqrt{2}$

\sqrt{2}

$\sqrt{2}$

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}}}

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

Step 9.2

Substitute the values of

b

$b$ and

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

$\sqrt{a^{2} + b^{2}}$ in the formula.

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

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

Step 9.3

Simplify.

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Step 9.3.1

One to any power is one.

\frac{1}{\sqrt{2}}

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

Step 9.3.2

Multiply

\frac{1}{\sqrt{2}}

$\frac{1}{\sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}

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

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

Multiply

\frac{1}{\sqrt{2}}

$\frac{1}{\sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}

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

Step 9.3.3.2

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

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

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

Step 9.3.3.3

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

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

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

Step 9.3.3.4

Use the power rule

a^{m} a^{n} = a^{m + n}

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

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

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

Step 9.3.3.5

Add

1

$1$ and

1

$1$ .

\frac{\sqrt{2}}{{\sqrt{2}}^{2}}

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

Step 9.3.3.6

Rewrite

{\sqrt{2}}^{2}

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

2

$2$ .

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Step 9.3.3.6.1

Use

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

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

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

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

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

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

Step 9.3.3.6.2

Apply the power rule and multiply exponents,

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

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

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

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

Step 9.3.3.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{\sqrt{2}}{2^{\frac{2}{2}}}

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

Step 9.3.3.6.4

Cancel the common factor of

2

$2$ .

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Step 9.3.3.6.4.1

Cancel the common factor.

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

Step 9.3.3.6.4.2

Rewrite the expression.

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

Step 9.3.3.6.5

Evaluate the exponent.

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

Step 10

The asymptotes follow the form

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

$y = \pm 1 \cdot x + 0$

Step 11

Simplify

$1 \cdot x + 0$ .

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Step 11.1

Add

$1 \cdot x$ and

$0$ .

$y = 1 \cdot x$

Step 11.2

Multiply

$x$ by

$1$ .

$y = x$

Step 12

Simplify

$- 1 \cdot x + 0$ .

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Step 12.1

Add

$- 1 \cdot x$ and

$0$ .

$y = - 1 \cdot x$

Step 12.2

Rewrite

$- 1 x$ as

$- x$ .

$y = - x$

Step 13

This hyperbola has two asymptotes.

$y = x, y = - x$

Step 14

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

Center:

$(0, 0)$

Vertices:

$(1, 0), (- 1, 0)$

Foci:

$(\sqrt{2}, 0), (- \sqrt{2}, 0)$

Eccentricity:

$\sqrt{2}$

Focal Parameter:

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

Asymptotes:

$y = x$ ,

$y = - x$

Step 15