Graph f(x,y)=x^2-y^2

$f (x, y) = x^{2} - y^{2}$

Step 1

Find the standard form of the hyperbola.

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

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

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

Add $y^{2}$ to both sides of the equation.

$y - x^{2} + y^{2} = 0$

Move $y$ .

$- x^{2} + y^{2} + y = 0$

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

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

$a = 1$

$b = 1$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{1}{2 \cdot 1}$

Cancel the common factor of $1$ .

Cancel the common factor.

$d = \frac{1}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{1}{2}$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{1^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

One to any power is one.

$e = 0 - \frac{1}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{1}{4}$

Subtract $\frac{1}{4}$ from $0$ .

$e = - \frac{1}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + \frac{1}{2})}^{2} - \frac{1}{4}$ .

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

Substitute ${(y + \frac{1}{2})}^{2} - \frac{1}{4}$ for $y^{2} + y$ in the equation $- x^{2} + y^{2} + y = 0$ .

$- x^{2} + {(y + \frac{1}{2})}^{2} - \frac{1}{4} = 0$

Move $- \frac{1}{4}$ to the right side of the equation by adding $\frac{1}{4}$ to both sides.

$- x^{2} + {(y + \frac{1}{2})}^{2} = 0 + \frac{1}{4}$

Add $0$ and $\frac{1}{4}$ .

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

Divide each term by $\frac{1}{4}$ to make the right side equal to one.

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

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 $1$ .

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

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, $a$ .

$a = \frac{1}{2}$

$b = \frac{1}{2}$

$k = - \frac{1}{2}$

$h = 0$

Step 4

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

$(0, - \frac{1}{2})$

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.

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

Simplify.

Simplify the expression.

Apply the product rule to $\frac{1}{2}$ .

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

One to any power is one.

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

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

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

Apply the product rule to $\frac{1}{2}$ .

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

One to any power is one.

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

$\sqrt{\frac{2}{4}}$

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

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Any root of $1$ is $1$ .

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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.

$(0, 0)$

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.

$(0, - 1)$

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

$(0, 0), (0, - 1)$

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.

$(0, - \frac{1}{2} + \frac{\sqrt{2}}{2})$

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.

$(0, - \frac{1}{2} - \frac{\sqrt{2}}{2})$

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

$(0, - \frac{1}{2} + \frac{\sqrt{2}}{2}), (0, - \frac{1}{2} - \frac{\sqrt{2}}{2})$

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.

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

Simplify.

Multiply the numerator by the reciprocal of the denominator.

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

Simplify the expression.

Apply the product rule to $\frac{1}{2}$ .

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

One to any power is one.

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

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

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

Apply the product rule to $\frac{1}{2}$ .

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

One to any power is one.

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

$\sqrt{\frac{2}{4}} \cdot 2$

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

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Any root of $1$ is $1$ .

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\sqrt{2}$

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.

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

Simplify.

Multiply the numerator by the reciprocal of the denominator.

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

Combine.

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

Simplify the expression.

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

$\frac{\frac{1}{4} \cdot 1}{\sqrt{2} \cdot 2}$

Multiply $\frac{1}{4}$ by $1$ .

$\frac{\frac{1}{4}}{\sqrt{2} \cdot 2}$

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

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

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{4} \cdot \frac{1}{2 \sqrt{2}}$

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

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

Combine and simplify the denominator.

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

$\frac{1}{4} \cdot \frac{\sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Move $\sqrt{2}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$\frac{1}{4} \cdot \frac{\sqrt{2}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\frac{1}{4} \cdot \frac{\sqrt{2}}{4}$

Multiply $\frac{1}{4} \cdot \frac{\sqrt{2}}{4}$ .

Multiply $\frac{1}{4}$ by $\frac{\sqrt{2}}{4}$ .

$\frac{\sqrt{2}}{4 \cdot 4}$

Multiply $4$ by $4$ .

$\frac{\sqrt{2}}{16}$

Step 10

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

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

Step 11

Multiply $x$ by $1$ .

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

Step 12

Rewrite $- 1 x$ as $- x$ .

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

Step 13

This hyperbola has two asymptotes.

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

Step 14

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

Center: $(0, - \frac{1}{2})$

Vertices: $(0, 0), (0, - 1)$

Foci: $(0, - \frac{1}{2} + \frac{\sqrt{2}}{2}), (0, - \frac{1}{2} - \frac{\sqrt{2}}{2})$

Eccentricity: $\sqrt{2}$

Focal Parameter: $\frac{\sqrt{2}}{16}$

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

Step 15

Graph f(x,y)=x^2-y^2

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