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# Trigonometry Examples

Divide each term by to make the right side equal to one.

Simplify each term in the equation in order to set the right side equal to . The standard form of an ellipse or hyperbola requires the right side of the equation be .

This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.

Match the values in this hyperbola to those of the standard form. The variable represents the x-offset from the origin, represents the y-offset from origin, .

The center of a hyperbola follows the form of . Substitute in the values of and .

Find the distance from the center to a focus of the hyperbola by using the following formula.

Substitute the values of and in the formula.

Simplify.

Remove parentheses.

Raise to the power of .

Remove parentheses.

Raise to the power of .

Add and .

Rewrite as .

Factor out of .

Rewrite as .

Pull terms out from under the radical.

The first vertex of a hyperbola can be found by adding to .

Substitute the known values of , , and into the formula and simplify.

The second vertex of a hyperbola can be found by subtracting from .

Substitute the known values of , , and into the formula and simplify.

The vertices of a hyperbola follow the form of . Hyperbolas have two vertices.

The first focus of a hyperbola can be found by adding to .

Substitute the known values of , , and into the formula and simplify.

The second focus of a hyperbola can be found by subtracting from .

Substitute the known values of , , and into the formula and simplify.

The foci of a hyperbola follow the form of . Hyperbolas have two foci.

Find the value of the eccentricity of the hyperbola by using the following formula.

Substitute the values of and in the formula.

Simplify.

Simplify the numerator.

Remove parentheses.

Raise to the power of .

Remove parentheses.

Raise to the power of .

Add and .

Rewrite as .

Factor out of .

Rewrite as .

Pull terms out from under the radical.

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by .

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

Substitute the values of and in the formula.

Simplify.

Raise to the power of .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Multiply by .

Simplify.

Combine.

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite as .

The asymptotes follow the form because this hyperbola opens left and right.

Multiply by .

Add and .

Multiply by .

Add and .

This hyperbola has two asymptotes.

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

Center:

Vertices:

Foci:

Eccentricity:

Focal Parameter:

Asymptotes: ,