Graph 3x^2-2y^2+4y-26=0

$3 x^{2} - 2 y^{2} + 4 y - 26 = 0$

Step 1

Find the standard form of the hyperbola.

Add $26$ to both sides of the equation.

$3 x^{2} - 2 y^{2} + 4 y = 26$

Complete the square for $- 2 y^{2} + 4 y$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 2$

$b = 4$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{4}{2 \cdot - 2}$

Simplify the right side.

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$d = \frac{2 \cdot 2}{2 \cdot - 2}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 2$ .

$d = \frac{2 \cdot 2}{2 (- 2)}$

Cancel the common factor.

$d = \frac{2 \cdot 2}{2 \cdot - 2}$

Rewrite the expression.

$d = \frac{2}{- 2}$

Cancel the common factor of $2$ and $- 2$ .

Factor $2$ out of $2$ .

$d = \frac{2 (1)}{- 2}$

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$d = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$d = - 1$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{4^{2}}{4 \cdot - 2}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $4^{2}$ and $4$ .

Factor $4$ out of $4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 \cdot - 2}$

Cancel the common factors.

Factor $4$ out of $4 \cdot - 2$ .

$e = 0 - \frac{4 \cdot 4}{4 (- 2)}$

Cancel the common factor.

$e = 0 - \frac{4 \cdot 4}{4 \cdot - 2}$

Rewrite the expression.

$e = 0 - \frac{4}{- 2}$

Cancel the common factor of $4$ and $- 2$ .

Factor $2$ out of $4$ .

$e = 0 - \frac{2 (2)}{- 2}$

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$e = 0 - (- 1 \cdot 2)$

Multiply $- (- 1 \cdot 2)$ .

Multiply $- 1$ by $2$ .

$e = 0 - - 2$

Multiply $- 1$ by $- 2$ .

$e = 0 + 2$

Add $0$ and $2$ .

$e = 2$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(y - 1)}^{2} + 2$ .

$- 2 {(y - 1)}^{2} + 2$

Substitute $- 2 {(y - 1)}^{2} + 2$ for $- 2 y^{2} + 4 y$ in the equation $3 x^{2} - 2 y^{2} + 4 y = 26$ .

$3 x^{2} - 2 {(y - 1)}^{2} + 2 = 26$

Move $2$ to the right side of the equation by adding $2$ to both sides.

$3 x^{2} - 2 {(y - 1)}^{2} = 26 - 2$

Subtract $2$ from $26$ .

$3 x^{2} - 2 {(y - 1)}^{2} = 24$

Divide each term by $24$ to make the right side equal to one.

$\frac{3 x^{2}}{24} - \frac{2 {(y - 1)}^{2}}{24} = \frac{24}{24}$

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 $1$ .

$\frac{x^{2}}{8} - \frac{{(y - 1)}^{2}}{12} = 1$

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, $a$ .

$a = 2 \sqrt{2}$

$b = 2 \sqrt{3}$

$k = 1$

$h = 0$

Step 4

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

$(0, 1)$

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.

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

Simplify.

Simplify the expression.

Apply the product rule to $2 \sqrt{2}$ .

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

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

$\sqrt{4 {\sqrt{2}}^{2} + {(2 \sqrt{3})}^{2}}$

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

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

$\sqrt{4 {(2^{\frac{1}{2}})}^{2} + {(2 \sqrt{3})}^{2}}$

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

$\sqrt{4 \cdot 2^{\frac{1}{2} \cdot 2} + {(2 \sqrt{3})}^{2}}$

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

$\sqrt{4 \cdot 2^{\frac{2}{2}} + {(2 \sqrt{3})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{4 \cdot 2^{\frac{2}{2}} + {(2 \sqrt{3})}^{2}}$

Rewrite the expression.

$\sqrt{4 \cdot 2^{1} + {(2 \sqrt{3})}^{2}}$

Evaluate the exponent.

$\sqrt{4 \cdot 2 + {(2 \sqrt{3})}^{2}}$

Simplify the expression.

Multiply $4$ by $2$ .

$\sqrt{8 + {(2 \sqrt{3})}^{2}}$

Apply the product rule to $2 \sqrt{3}$ .

$\sqrt{8 + 2^{2} {\sqrt{3}}^{2}}$

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

$\sqrt{8 + 4 {\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{8 + 4 {(3^{\frac{1}{2}})}^{2}}$

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

$\sqrt{8 + 4 \cdot 3^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{8 + 4 \cdot 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{8 + 4 \cdot 3^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{8 + 4 \cdot 3^{1}}$

Evaluate the exponent.

$\sqrt{8 + 4 \cdot 3}$

Simplify the expression.

Multiply $4$ by $3$ .

$\sqrt{8 + 12}$

Add $8$ and $12$ .

$\sqrt{20}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$\sqrt{4 (5)}$

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

$2 \sqrt{5}$

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.

$(2 \sqrt{2}, 1)$

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.

$(- 2 \sqrt{2}, 1)$

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

$(2 \sqrt{2}, 1), (- 2 \sqrt{2}, 1)$

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.

$(2 \sqrt{5}, 1)$

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.

$(- 2 \sqrt{5}, 1)$

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

$(2 \sqrt{5}, 1), (- 2 \sqrt{5}, 1)$

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.

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

Simplify.

Simplify the numerator.

Apply the product rule to $2 \sqrt{2}$ .

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

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

$\frac{\sqrt{4 {\sqrt{2}}^{2} + {(2 \sqrt{3})}^{2}}}{2 \sqrt{2}}$

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

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

$\frac{\sqrt{4 {(2^{\frac{1}{2}})}^{2} + {(2 \sqrt{3})}^{2}}}{2 \sqrt{2}}$

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

$\frac{\sqrt{4 \cdot 2^{\frac{1}{2} \cdot 2} + {(2 \sqrt{3})}^{2}}}{2 \sqrt{2}}$

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

$\frac{\sqrt{4 \cdot 2^{\frac{2}{2}} + {(2 \sqrt{3})}^{2}}}{2 \sqrt{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{4 \cdot 2^{\frac{2}{2}} + {(2 \sqrt{3})}^{2}}}{2 \sqrt{2}}$

Rewrite the expression.

$\frac{\sqrt{4 \cdot 2^{1} + {(2 \sqrt{3})}^{2}}}{2 \sqrt{2}}$

Evaluate the exponent.

$\frac{\sqrt{4 \cdot 2 + {(2 \sqrt{3})}^{2}}}{2 \sqrt{2}}$

Multiply $4$ by $2$ .

$\frac{\sqrt{8 + {(2 \sqrt{3})}^{2}}}{2 \sqrt{2}}$

Apply the product rule to $2 \sqrt{3}$ .

$\frac{\sqrt{8 + 2^{2} {\sqrt{3}}^{2}}}{2 \sqrt{2}}$

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

$\frac{\sqrt{8 + 4 {\sqrt{3}}^{2}}}{2 \sqrt{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{8 + 4 {(3^{\frac{1}{2}})}^{2}}}{2 \sqrt{2}}$

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

$\frac{\sqrt{8 + 4 \cdot 3^{\frac{1}{2} \cdot 2}}}{2 \sqrt{2}}$

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

$\frac{\sqrt{8 + 4 \cdot 3^{\frac{2}{2}}}}{2 \sqrt{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{8 + 4 \cdot 3^{\frac{2}{2}}}}{2 \sqrt{2}}$

Rewrite the expression.

$\frac{\sqrt{8 + 4 \cdot 3^{1}}}{2 \sqrt{2}}$

Evaluate the exponent.

$\frac{\sqrt{8 + 4 \cdot 3}}{2 \sqrt{2}}$

Multiply $4$ by $3$ .

$\frac{\sqrt{8 + 12}}{2 \sqrt{2}}$

Add $8$ and $12$ .

$\frac{\sqrt{20}}{2 \sqrt{2}}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$\frac{\sqrt{4 (5)}}{2 \sqrt{2}}$

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{\sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{\sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

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

$\frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $5$ by $2$ .

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

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.

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

Simplify.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{{(3^{\frac{1}{2}})}^{2}}{\sqrt{5}}$

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

$\frac{3^{\frac{1}{2} \cdot 2}}{\sqrt{5}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{3^{1}}{\sqrt{5}}$

Evaluate the exponent.

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

Multiply $\frac{3}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{3}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

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

$\frac{3 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

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

$\frac{3 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

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

$\frac{3 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

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

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

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

$\frac{3 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

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

$\frac{3 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{3 \sqrt{5}}{5^{1}}$

Evaluate the exponent.

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

Step 10

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

$y = \pm \frac{\sqrt{6}}{2} x + 1$

Step 11

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

$y = \frac{\sqrt{6} x}{2} + 1$

Step 12

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

$y = - \frac{x \sqrt{6}}{2} + 1$

Step 13

This hyperbola has two asymptotes.

$y = \frac{\sqrt{6} x}{2} + 1, y = - \frac{x \sqrt{6}}{2} + 1$

Step 14

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

Center: $(0, 1)$

Vertices: $(2 \sqrt{2}, 1), (- 2 \sqrt{2}, 1)$

Foci: $(2 \sqrt{5}, 1), (- 2 \sqrt{5}, 1)$

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

Focal Parameter: $\frac{3 \sqrt{5}}{5}$

Asymptotes: $y = \frac{\sqrt{6} x}{2} + 1$ , $y = - \frac{x \sqrt{6}}{2} + 1$

Step 15

Graph 3x^2-2y^2+4y-26=0

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