Precalculus Examples

$\frac{{(x - 1)}^{2}}{8^{2}} - \frac{{(y - 11)}^{2}}{7^{2}} = 1$

Step 1

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 $1$ .

$\frac{{(x - 1)}^{2}}{64} - \frac{{(y - 11)}^{2}}{49} = 1$

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.

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

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, $a$ .

$a = 8$

$b = 7$

$k = 11$

$h = 1$

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 $h$ .

$(h + a, k)$

Step 4.2

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

$(9, 11)$

Step 4.3

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

$(h - a, k)$

Step 4.4

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

$(- 7, 11)$

Step 4.5

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

$(9, 11), (- 7, 11)$

Step 5