Graph x^2-y^2-2x=0 - MathMaster

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

Find the standard form of the hyperbola.

Tap for more steps…

Complete the square for $x^{2} - 2 x$ .

Tap for more steps…

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

$a = 1, b = - 2, c = 0$

Consider the vertex form of a parabola.

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

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

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

Simplify the right side.

Tap for more steps…

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

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

Rewrite the expression.

$d = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$d = - 1$

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

Tap for more steps…

Simplify each term.

Tap for more steps…

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

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

Multiply $4$ by $1$ .

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

Cancel the common factor of $4$ .

Tap for more steps…

Cancel the common factor.

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

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Subtract $1$ from $0$ .

$e = - 1$

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

${(x - 1)}^{2} - 1$

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

${(x - 1)}^{2} - 1 - y^{2} = 0$

Move $- 1$ to the right side of the equation by adding $1$ to both sides.

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

Add $0$ and $1$ .

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

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 $1$ .

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

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, $a$ .

$a = 1$

$b = 1$

$k = 0$

$h = 1$

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

$(1, 0)$

Find $c$ , the distance from the center to a focus.

Tap for more steps…

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.

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

Simplify.

Tap for more steps…

One to any power is one.

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

One to any power is one.

$\sqrt{1 + 1}$

Add $1$ and $1$ .

$\sqrt{2}$

Find the vertices.

Tap for more steps…

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.

$(2, 0)$

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.

$(0, 0)$

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

$(2, 0), (0, 0)$

Find the foci.

Tap for more steps…

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.

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

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.

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

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

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

Find the eccentricity.

Tap for more steps…

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.

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

Simplify.

Tap for more steps…

Divide $\sqrt{{(1)}^{2} + {(1)}^{2}}$ by $1$ .

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

One to any power is one.

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

One to any power is one.

$\sqrt{1 + 1}$

Add $1$ and $1$ .

$\sqrt{2}$

Find the focal parameter.

Tap for more steps…

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.

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

Simplify.

Tap for more steps…

One to any power is one.

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

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.

Tap for more steps…

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

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

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 $2$ .

Tap for more steps…

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 $2$ .

Tap for more steps…

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

$y = \pm 1 \cdot (x - (1)) + 0$

Simplify to find the first asymptote.

Tap for more steps…

Remove parentheses.

$y = 1 \cdot (x - (1)) + 0$

Simplify $1 \cdot (x - (1)) + 0$ .

Tap for more steps…

Add $1 \cdot (x - (1))$ and $0$ .

$y = 1 \cdot (x - (1))$

Multiply $x - (1)$ by $1$ .

$y = x - (1)$

Multiply $- 1$ by $1$ .

$y = x - 1$

Simplify to find the second asymptote.

Tap for more steps…

Remove parentheses.

$y = - 1 \cdot (x - (1)) + 0$

Simplify $- 1 \cdot (x - (1)) + 0$ .

Tap for more steps…

Simplify the expression.

Tap for more steps…

Add $- 1 \cdot (x - (1))$ and $0$ .

$y = - 1 \cdot (x - (1))$

Multiply $- 1$ by $1$ .

$y = - 1 \cdot (x - 1)$

Apply the distributive property.

$y = - 1 x - 1 \cdot - 1$

Simplify the expression.

Tap for more steps…

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

$y = - x - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$y = - x + 1$

This hyperbola has two asymptotes.

$y = x - 1, y = - x + 1$

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

Center: $(1, 0)$

Vertices: $(2, 0), (0, 0)$

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

Eccentricity: $\sqrt{2}$

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

Asymptotes: $y = x - 1$ , $y = - x + 1$

Graph x^2-y^2-2x=0

Download our
App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our
App from the store