Graph y^2=2x^2-1

$y^{2} = 2 x^{2} - 1$

Find the standard form of the hyperbola.

Move all terms containing variables to the left side of the equation.

Subtract $2 x^{2}$ from both sides of the equation.

$y^{2} - 2 x^{2} = - 1$

Reorder $y^{2}$ and $- 2 x^{2}$ .

$- 2 x^{2} + y^{2} = - 1$

Flip the sign on each term of the equation so the term on the right side is positive.

$2 x^{2} - y^{2} = 1$

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 $1$ .

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

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, $a$ .

$a = \frac{\sqrt{2}}{2}$

$b = 1$

$k = 0$

$h = 0$

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

$(0, 0)$

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.

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

Simplify.

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the expression.

One to any power is one.

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $1$ and $2$ .

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

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

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

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

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

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

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

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

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

$\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

$\frac{\sqrt{6}}{2}$

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.

$(\frac{\sqrt{2}}{2}, 0)$

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.

$(- \frac{\sqrt{2}}{2}, 0)$

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

$(\frac{\sqrt{2}}{2}, 0), (- \frac{\sqrt{2}}{2}, 0)$

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.

$(\frac{\sqrt{6}}{2}, 0)$

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.

$(- \frac{\sqrt{6}}{2}, 0)$

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

$(\frac{\sqrt{6}}{2}, 0), (- \frac{\sqrt{6}}{2}, 0)$

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.

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

Simplify.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

$\sqrt{\frac{2}{4} + 1^{2}} \frac{2}{\sqrt{2}}$

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

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the expression.

One to any power is one.

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $1$ and $2$ .

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

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

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

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Cancel the common factor of $\sqrt{2}$ .

Factor $\sqrt{2}$ out of $\sqrt{3} \sqrt{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\sqrt{3}$

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.

$\frac{\frac{1^{2}}{\sqrt{6}}}{2}$

Simplify.

Multiply the numerator by the reciprocal of the denominator.

$\frac{1^{2}}{\sqrt{6}} \cdot \frac{1}{2}$

Combine.

$\frac{1^{2} \cdot 1}{\sqrt{6} \cdot 2}$

Multiply $1^{2}$ by $1$ by adding the exponents.

Multiply $1^{2}$ by $1$ .

Raise $1$ to the power of $1$ .

$\frac{1^{2} \cdot 1^{1}}{\sqrt{6} \cdot 2}$

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

$\frac{1^{2 + 1}}{\sqrt{6} \cdot 2}$

Add $2$ and $1$ .

$\frac{1^{3}}{\sqrt{6} \cdot 2}$

One to any power is one.

$\frac{1}{\sqrt{6} \cdot 2}$

Move $2$ to the left of $\sqrt{6}$ .

$\frac{1}{2 \sqrt{6}}$

Multiply $\frac{1}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{1}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Combine and simplify the denominator.

Multiply $\frac{1}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

Move $\sqrt{6}$ .

$\frac{\sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{6}}{2 ({\sqrt{6}}^{1} \sqrt{6})}$

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{6}}{2 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}$

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

$\frac{\sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{\sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

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

$\frac{\sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt{6}}{2 \cdot 6^{1}}$

Evaluate the exponent.

$\frac{\sqrt{6}}{2 \cdot 6}$

Multiply $2$ by $6$ .

$\frac{\sqrt{6}}{12}$

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

$y = \pm \sqrt{2} \cdot x + 0$

Add $\sqrt{2} \cdot x$ and $0$ .

$y = \sqrt{2} x$

Add $- \sqrt{2} \cdot x$ and $0$ .

$y = - \sqrt{2} x$

This hyperbola has two asymptotes.

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

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

Center: $(0, 0)$

Vertices: $(\frac{\sqrt{2}}{2}, 0), (- \frac{\sqrt{2}}{2}, 0)$

Foci: $(\frac{\sqrt{6}}{2}, 0), (- \frac{\sqrt{6}}{2}, 0)$

Eccentricity: $\sqrt{3}$

Focal Parameter: $\frac{\sqrt{6}}{12}$

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

Graph y^2=2x^2-1

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