Graph (y^2)/9-(x^2)/36=1

$\frac{y^{2}}{9} - \frac{x^{2}}{36} = 1$

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 $1$ .

$\frac{y^{2}}{9} - \frac{x^{2}}{36} = 1$

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, $a$ .

$a = 3$

$b = 6$

$k = 0$

$h = 0$

Step 4

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

$(0, 0)$

Step 5

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.

$\sqrt{{(3)}^{2} + {(6)}^{2}}$

Simplify.

Raise $3$ to the power of $2$ .

$\sqrt{9 + {(6)}^{2}}$

Raise $6$ to the power of $2$ .

$\sqrt{9 + 36}$

Add $9$ and $36$ .

$\sqrt{45}$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$\sqrt{9 (5)}$

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2} \cdot 5}$

Pull terms out from under the radical.

$3 \sqrt{5}$

Step 6

Find the vertices.

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

$(h, k + a)$

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

$(0, 3)$

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

$(h, k - a)$

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

$(0, - 3)$

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

$(0, 3), (0, - 3)$

Step 7

Find the foci.

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

$(h, k + c)$

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

$(0, 3 \sqrt{5})$

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

$(h, k - c)$

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

$(0, - 3 \sqrt{5})$

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

$(0, 3 \sqrt{5}), (0, - 3 \sqrt{5})$

Step 8

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.

$\frac{\sqrt{{(3)}^{2} + {(6)}^{2}}}{3}$

Simplify.

Simplify the numerator.

Raise $3$ to the power of $2$ .

$\frac{\sqrt{9 + 6^{2}}}{3}$

Raise $6$ to the power of $2$ .

$\frac{\sqrt{9 + 36}}{3}$

Add $9$ and $36$ .

$\frac{\sqrt{45}}{3}$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$\frac{\sqrt{9 (5)}}{3}$

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 5}}{3}$

Pull terms out from under the radical.

$\frac{3 \sqrt{5}}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 \sqrt{5}}{3}$

Divide $\sqrt{5}$ by $1$ .

$\sqrt{5}$

Step 9

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.

$\frac{6^{2}}{3 \sqrt{5}}$

Simplify.

Raise $6$ to the power of $2$ .

$\frac{36}{3 \sqrt{5}}$

Cancel the common factor of $36$ and $3$ .

Factor $3$ out of $36$ .

$\frac{3 \cdot 12}{3 \sqrt{5}}$

Cancel the common factors.

Factor $3$ out of $3 \sqrt{5}$ .

$\frac{3 \cdot 12}{3 (\sqrt{5})}$

Cancel the common factor.

$\frac{3 \cdot 12}{3 \sqrt{5}}$

Rewrite the expression.

$\frac{12}{\sqrt{5}}$

Multiply $\frac{12}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{12}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Combine and simplify the denominator.

Multiply $\frac{12}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{12 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{12 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{12 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

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

$\frac{12 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{12 \sqrt{5}}{{\sqrt{5}}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{12 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

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

$\frac{12 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

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

$\frac{12 \sqrt{5}}{5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{12 \sqrt{5}}{5^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{12 \sqrt{5}}{5^{1}}$

Evaluate the exponent.

$\frac{12 \sqrt{5}}{5}$

Step 10

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

$y = \pm \frac{1}{2} x + 0$

Step 11

Simplify $\frac{1}{2} x + 0$ .

Add $\frac{1}{2} x$ and $0$ .

$y = \frac{1}{2} x$

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

$y = \frac{x}{2}$

Step 12

Simplify $- \frac{1}{2} x + 0$ .

Add $- \frac{1}{2} x$ and $0$ .

$y = - \frac{1}{2} x$

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

$y = - \frac{x}{2}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{x}{2}, y = - \frac{x}{2}$

Step 14

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

Center: $(0, 0)$

Vertices: $(0, 3), (0, - 3)$

Foci: $(0, 3 \sqrt{5}), (0, - 3 \sqrt{5})$

Eccentricity: $\sqrt{5}$

Focal Parameter: $\frac{12 \sqrt{5}}{5}$

Asymptotes: $y = \frac{x}{2}$ , $y = - \frac{x}{2}$

Step 15

Graph (y^2)/9-(x^2)/36=1

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