Trigonometry Examples

f (x) = x^{2} + c

$f (x) = x^{2} + c$

Step 1

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

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Step 1.1

Subtract

x^{2}

$x^{2}$ from both sides of the equation.

$y - x^{2} = c$

Step 1.2

Subtract

$c$ from both sides of the equation.

$y - x^{2} - c = 0$

Step 1.3

Move

$y$ .

$- x^{2} - c + y = 0$

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,

$a$ .

$a = 1$

$b = 1$

$k = 0$

$h = 0$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

One to any power is one.

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

Step 5.3.2

One to any power is one.

$\sqrt{1 + 1}$

Step 5.3.3

Add

$1$ and

$1$ .

$\sqrt{2}$

Step 6

Find the vertices.

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Step 6.1

The first vertex of a hyperbola can be found by adding

$a$ to

$k$ .

$(h, k + a)$

Step 6.2

Substitute the known values of

$h$ ,

$a$ , and

$k$ into the formula and simplify.

$(0, 1)$

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.

$(0, - 1)$

Step 6.5

The vertices of a hyperbola follow the form of

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

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

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.

$(0, \sqrt{2})$

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.

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

Step 7.5

The foci of a hyperbola follow the form of

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

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

Step 8

Find the focal parameter.

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Step 8.1

Find the value of the focal parameter of the hyperbola by using the following formula.

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

Step 8.2

Substitute the values of

$b$ and

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

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

Step 8.3

Simplify.

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Step 8.3.1

One to any power is one.

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

Step 8.3.2

Multiply

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

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

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

Step 8.3.3

Combine and simplify the denominator.

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Step 8.3.3.1

Multiply

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

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

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

Step 8.3.3.2

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 8.3.3.3

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 8.3.3.4

Use the power rule

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

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

Step 8.3.3.5

Add

$1$ and

$1$ .

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

Step 8.3.3.6

Rewrite

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

$2$ .

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Step 8.3.3.6.1

Use

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

$\sqrt{2}$ as

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

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

Step 8.3.3.6.2

Apply the power rule and multiply exponents,

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

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

Step 8.3.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 8.3.3.6.4

Cancel the common factor of

$2$ .

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Step 8.3.3.6.4.1

Cancel the common factor.

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

Step 8.3.3.6.4.2

Rewrite the expression.

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

Step 8.3.3.6.5

Evaluate the exponent.

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

Step 9

The asymptotes follow the form

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

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

Step 10

Simplify

$1 \cdot x + 0$ .

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Step 10.1

Add

$1 \cdot x$ and

$0$ .

$y = 1 \cdot x$

Step 10.2

Multiply

$x$ by

$1$ .

$y = x$

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

Rewrite

$- 1 x$ as

$- x$ .

$y = - x$

Step 12

This hyperbola has two asymptotes.

$y = x, y = - x$

Step 13

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

Center:

$(0, 0)$

Vertices:

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

Foci:

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

Eccentricity:

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

Focal Parameter:

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

Asymptotes:

$y = x$ ,

$y = - x$

Step 14