Precalculus Examples

$4 {(y - 1)}^{2} - 9 {(x - 3)}^{2} = 36$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Divide each term by $36$ to make the right side equal to one.

$\frac{4 {(y - 1)}^{2}}{36} - \frac{9 {(x - 3)}^{2}}{36} = \frac{36}{36}$

Step 1.2

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{{(y - 1)}^{2}}{9} - \frac{{(x - 3)}^{2}}{4} = 1$

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.

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

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, $a$ .

$a = 3$

$b = 2$

$k = 1$

$h = 3$

Step 4

Find the vertices.

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Step 4.1

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

$(h, k + a)$

Step 4.2

Substitute the known values of $h$ , $a$ , and $k$ into the formula and simplify.

$(3, 4)$

Step 4.3

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

$(h, k - a)$

Step 4.4

Substitute the known values of $h$ , $a$ , and $k$ into the formula and simplify.

$(3, - 2)$

Step 4.5

The vertices of a hyperbola follow the form of $(h, k \pm a)$ . Hyperbolas have two vertices.

$(3, 4), (3, - 2)$

Step 5