Graph ((x-h)^2)/(a^2)-((y-k)^2)/(b^2)=1

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

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 $1$ .

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

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

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

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, $a$ .

$a = a$

$b = b$

$k = k$

$h = h$

The center of a hyperbola follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(h, k)$

Find $c$ , the distance from the center to a focus.

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

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

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(a)}^{2} + {(b)}^{2}}$

Remove parentheses.

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

Find the vertices.

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

$(h + a, k)$

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

$(h + a, k)$

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

$(h - a, k)$

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

$(h - a, k)$

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

$(h + a, k), (h - a, k)$

Find the foci.

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

$(h + c, k)$

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

$(h + \sqrt{a^{2} + b^{2}}, k)$

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

$(h - c, k)$

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

$(h - \sqrt{a^{2} + b^{2}}, k)$

The foci of a hyperbola follow the form of $(h \pm \sqrt{a^{2} + b^{2}}, k)$ . Hyperbolas have two foci.

$(h + \sqrt{a^{2} + b^{2}}, k), (h - \sqrt{a^{2} + b^{2}}, k)$

Find the eccentricity.

Find the eccentricity by using the following formula.

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

Substitute the values of $a$ and $b$ into the formula.

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

Remove parentheses.

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

Find the focal parameter.

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

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

Substitute the values of $b$ and $\sqrt{a^{2} + b^{2}}$ in the formula.

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

Simplify.

Multiply $\frac{b^{2}}{\sqrt{a^{2} + b^{2}}}$ by $\frac{\sqrt{a^{2} + b^{2}}}{\sqrt{a^{2} + b^{2}}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{b^{2}}{\sqrt{a^{2} + b^{2}}}$ by $\frac{\sqrt{a^{2} + b^{2}}}{\sqrt{a^{2} + b^{2}}}$ .

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

Raise $\sqrt{a^{2} + b^{2}}$ to the power of $1$ .

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

Raise $\sqrt{a^{2} + b^{2}}$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{a^{2} + b^{2}}}^{2}$ as $a^{2} + b^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a^{2} + b^{2}}$ as ${(a^{2} + b^{2})}^{\frac{1}{2}}$ .

$\frac{b^{2} \sqrt{a^{2} + b^{2}}}{{({(a^{2} + b^{2})}^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{b^{2} \sqrt{a^{2} + b^{2}}}{{(a^{2} + b^{2})}^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{b^{2} \sqrt{a^{2} + b^{2}}}{{(a^{2} + b^{2})}^{1}}$

Simplify.

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

The asymptotes follow the form $y = \pm \frac{b (x - h)}{a} + k$ because this hyperbola opens left and right.

$y = \pm \frac{b}{a} \cdot (x - (h)) + k$

Simplify to find the first asymptote.

Multiply $- 1$ by $h$ .

$y = \frac{b}{a} \cdot (x - h) + k$

Remove parentheses.

$y = \frac{b}{a} \cdot (x - (h)) + k$

Simplify $\frac{b}{a} \cdot (x - (h)) + k$ .

Multiply $\frac{b}{a}$ by $x - h$ .

$y = \frac{b (x - h)}{a} + k$

To write $k$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$y = \frac{b (x - h)}{a} + \frac{k a}{a}$

Combine the numerators over the common denominator.

$y = \frac{b (x - h) + k a}{a}$

Simplify the numerator.

Apply the distributive property.

$y = \frac{b x + b (- h) + k a}{a}$

Rewrite using the commutative property of multiplication.

$y = \frac{b x - b h + k a}{a}$

Simplify to find the second asymptote.

Multiply $- 1$ by $h$ .

$y = - \frac{b}{a} \cdot (x - h) + k$

Remove parentheses.

$y = - \frac{b}{a} \cdot (x - (h)) + k$

Simplify $- \frac{b}{a} \cdot (x - (h)) + k$ .

Simplify each term.

Apply the distributive property.

$y = - \frac{b}{a} x - \frac{b}{a} (- h) + k$

Combine $x$ and $\frac{b}{a}$ .

$y = - \frac{x b}{a} - \frac{b}{a} (- h) + k$

Multiply $- \frac{b}{a} (- h)$ .

Multiply $- 1$ by $- 1$ .

$y = - \frac{x b}{a} + 1 \frac{b}{a} h + k$

Multiply $\frac{b}{a}$ by $1$ .

$y = - \frac{x b}{a} + \frac{b}{a} h + k$

Combine $\frac{b}{a}$ and $h$ .

$y = - \frac{x b}{a} + \frac{b h}{a} + k$

Simplify with commuting.

Reorder $x$ and $b$ .

$y = - \frac{b x}{a} + \frac{b h}{a} + k$

Reorder $- \frac{b x}{a}$ and $\frac{b h}{a}$ .

$y = \frac{b h}{a} - \frac{b x}{a} + k$

This hyperbola has two asymptotes.

$y = \frac{b x - b h + k a}{a}, y = \frac{b h}{a} - \frac{b x}{a} + k$

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

Center: $(h, k)$

Vertices: $(h + a, k), (h - a, k)$

Foci: $(h + \sqrt{a^{2} + b^{2}}, k), (h - \sqrt{a^{2} + b^{2}}, k)$

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

Focal Parameter: $\frac{b^{2} \sqrt{a^{2} + b^{2}}}{a^{2} + b^{2}}$

Asymptotes: $y = \frac{b x - b h + k a}{a}$ , $y = \frac{b h}{a} - \frac{b x}{a} + k$

Graph ((x-h)^2)/(a^2)-((y-k)^2)/(b^2)=1

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