Complete the square for .

Use the form , to find the values of , , and .

Consider the vertex form of a parabola.

Substitute the values of and into the formula .

Simplify the right side.

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Multiply by .

Find the value of using the formula .

Simplify each term.

Raise to the power of .

Multiply by .

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Multiply by .

Subtract from .

Substitute the values of , , and into the vertex form .

Substitute for in the equation .

Move to the right side of the equation by adding to both sides.

Add and .

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.

One to any power is one.

One to any power is one.

Add and .

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 eccentricity by using the following formula.

Substitute the values of and into the formula.

Simplify.

Divide by .

One to any power is one.

One to any power is one.

Add and .

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

Substitute the values of and in the formula.

Simplify.

One to any power is one.

Multiply by .

Combine and simplify the denominator.

Multiply and .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite as .

Use to rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Evaluate the exponent.

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

Remove parentheses.

Simplify .

Add and .

Multiply by .

Multiply by .

Remove parentheses.

Simplify .

Simplify the expression.

Add and .

Multiply by .

Apply the distributive property.

Simplify the expression.

Rewrite as .

Multiply by .

This hyperbola has two asymptotes.

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

Center:

Vertices:

Foci:

Eccentricity:

Focal Parameter:

Asymptotes: ,

Graph x^2-y^2-2x=0