Graph ((x-3)^2)/10-((y+4)^2)/15=1

$\frac{{(x - 3)}^{2}}{10} - \frac{{(y + 4)}^{2}}{15} = 1$

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 $1$ .

$\frac{{(x - 3)}^{2}}{10} - \frac{{(y + 4)}^{2}}{15} = 1$

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, $a$ .

$a = \sqrt{10}$

$b = \sqrt{15}$

$k = - 4$

$h = 3$

Step 4

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

$(3, - 4)$

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.

$\sqrt{{(\sqrt{10})}^{2} + {(\sqrt{15})}^{2}}$

Simplify.

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sqrt{{(10^{\frac{1}{2}})}^{2} + {(\sqrt{15})}^{2}}$

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

$\sqrt{10^{\frac{1}{2} \cdot 2} + {(\sqrt{15})}^{2}}$

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

$\sqrt{10^{\frac{2}{2}} + {(\sqrt{15})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{10^{\frac{2}{2}} + {(\sqrt{15})}^{2}}$

Rewrite the expression.

$\sqrt{10^{1} + {(\sqrt{15})}^{2}}$

Evaluate the exponent.

$\sqrt{10 + {(\sqrt{15})}^{2}}$

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\sqrt{10 + {(15^{\frac{1}{2}})}^{2}}$

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

$\sqrt{10 + 15^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{10 + 15^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{10 + 15^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{10 + 15^{1}}$

Evaluate the exponent.

$\sqrt{10 + 15}$

Simplify the expression.

Add $10$ and $15$ .

$\sqrt{25}$

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$5$

Step 6

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.

$(3 + \sqrt{10}, - 4)$

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.

$(3 - \sqrt{10}, - 4)$

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

$(3 + \sqrt{10}, - 4), (3 - \sqrt{10}, - 4)$

Step 7

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.

$(8, - 4)$

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.

$(- 2, - 4)$

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

$(8, - 4), (- 2, - 4)$

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.

$\frac{\sqrt{{(\sqrt{10})}^{2} + {(\sqrt{15})}^{2}}}{\sqrt{10}}$

Simplify.

Simplify the numerator.

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{\sqrt{{(10^{\frac{1}{2}})}^{2} + {\sqrt{15}}^{2}}}{\sqrt{10}}$

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

$\frac{\sqrt{10^{\frac{1}{2} \cdot 2} + {\sqrt{15}}^{2}}}{\sqrt{10}}$

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

$\frac{\sqrt{10^{\frac{2}{2}} + {\sqrt{15}}^{2}}}{\sqrt{10}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{10^{\frac{2}{2}} + {\sqrt{15}}^{2}}}{\sqrt{10}}$

Rewrite the expression.

$\frac{\sqrt{10^{1} + {\sqrt{15}}^{2}}}{\sqrt{10}}$

Evaluate the exponent.

$\frac{\sqrt{10 + {\sqrt{15}}^{2}}}{\sqrt{10}}$

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\frac{\sqrt{10 + {(15^{\frac{1}{2}})}^{2}}}{\sqrt{10}}$

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

$\frac{\sqrt{10 + 15^{\frac{1}{2} \cdot 2}}}{\sqrt{10}}$

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

$\frac{\sqrt{10 + 15^{\frac{2}{2}}}}{\sqrt{10}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{10 + 15^{\frac{2}{2}}}}{\sqrt{10}}$

Rewrite the expression.

$\frac{\sqrt{10 + 15^{1}}}{\sqrt{10}}$

Evaluate the exponent.

$\frac{\sqrt{10 + 15}}{\sqrt{10}}$

Add $10$ and $15$ .

$\frac{\sqrt{25}}{\sqrt{10}}$

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2}}}{\sqrt{10}}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{5}{\sqrt{10}}$

Multiply $\frac{5}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{5}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Combine and simplify the denominator.

Multiply $\frac{5}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{5 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{5 \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{5 \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

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

$\frac{5 \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{5 \sqrt{10}}{{\sqrt{10}}^{2}}$

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{5 \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

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

$\frac{5 \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

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

$\frac{5 \sqrt{10}}{10^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5 \sqrt{10}}{10^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{5 \sqrt{10}}{10^{1}}$

Evaluate the exponent.

$\frac{5 \sqrt{10}}{10}$

Cancel the common factor of $5$ and $10$ .

Factor $5$ out of $5 \sqrt{10}$ .

$\frac{5 (\sqrt{10})}{10}$

Cancel the common factors.

Factor $5$ out of $10$ .

$\frac{5 \sqrt{10}}{5 \cdot 2}$

Cancel the common factor.

$\frac{5 \sqrt{10}}{5 \cdot 2}$

Rewrite the expression.

$\frac{\sqrt{10}}{2}$

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.

$\frac{{\sqrt{15}}^{2}}{5}$

Simplify.

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\frac{{(15^{\frac{1}{2}})}^{2}}{5}$

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

$\frac{15^{\frac{1}{2} \cdot 2}}{5}$

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

$\frac{15^{\frac{2}{2}}}{5}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{15^{\frac{2}{2}}}{5}$

Rewrite the expression.

$\frac{15^{1}}{5}$

Evaluate the exponent.

$\frac{15}{5}$

Divide $15$ by $5$ .

$3$

Step 10

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

$y = \pm \frac{\sqrt{6}}{2} \cdot (x - (3)) - 4$

Step 11

Simplify to find the first asymptote.

Remove parentheses.

$y = \frac{\sqrt{6}}{2} \cdot (x - (3)) - 4$

Simplify each term.

Multiply $- 1$ by $3$ .

$y = \frac{\sqrt{6}}{2} \cdot (x - 3) - 4$

Apply the distributive property.

$y = \frac{\sqrt{6}}{2} x + \frac{\sqrt{6}}{2} \cdot - 3 - 4$

Combine $\frac{\sqrt{6}}{2}$ and $x$ .

$y = \frac{\sqrt{6} x}{2} + \frac{\sqrt{6}}{2} \cdot - 3 - 4$

Combine $\frac{\sqrt{6}}{2}$ and $- 3$ .

$y = \frac{\sqrt{6} x}{2} + \frac{\sqrt{6} \cdot - 3}{2} - 4$

Simplify each term.

Move $- 3$ to the left of $\sqrt{6}$ .

$y = \frac{\sqrt{6} x}{2} + \frac{- 3 \sqrt{6}}{2} - 4$

Move the negative in front of the fraction.

$y = \frac{\sqrt{6} x}{2} - \frac{3 \sqrt{6}}{2} - 4$

Step 12

Simplify to find the second asymptote.

Remove parentheses.

$y = - \frac{\sqrt{6}}{2} \cdot (x - (3)) - 4$

Simplify each term.

Multiply $- 1$ by $3$ .

$y = - \frac{\sqrt{6}}{2} \cdot (x - 3) - 4$

Apply the distributive property.

$y = - \frac{\sqrt{6}}{2} x - \frac{\sqrt{6}}{2} \cdot - 3 - 4$

Combine $x$ and $\frac{\sqrt{6}}{2}$ .

$y = - \frac{x \sqrt{6}}{2} - \frac{\sqrt{6}}{2} \cdot - 3 - 4$

Multiply $- \frac{\sqrt{6}}{2} \cdot - 3$ .

Multiply $- 3$ by $- 1$ .

$y = - \frac{x \sqrt{6}}{2} + 3 \frac{\sqrt{6}}{2} - 4$

Combine $3$ and $\frac{\sqrt{6}}{2}$ .

$y = - \frac{x \sqrt{6}}{2} + \frac{3 \sqrt{6}}{2} - 4$

Step 13

This hyperbola has two asymptotes.

$y = \frac{\sqrt{6} x}{2} - \frac{3 \sqrt{6}}{2} - 4, y = - \frac{x \sqrt{6}}{2} + \frac{3 \sqrt{6}}{2} - 4$

Step 14

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

Center: $(3, - 4)$

Vertices: $(3 + \sqrt{10}, - 4), (3 - \sqrt{10}, - 4)$

Foci: $(8, - 4), (- 2, - 4)$

Eccentricity: $\frac{\sqrt{10}}{2}$

Focal Parameter: $3$

Asymptotes: $y = \frac{\sqrt{6} x}{2} - \frac{3 \sqrt{6}}{2} - 4$ , $y = - \frac{x \sqrt{6}}{2} + \frac{3 \sqrt{6}}{2} - 4$

Step 15

Graph ((x-3)^2)/10-((y+4)^2)/15=1

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