Graph ((x+5)^2)/4-y^2=1

$\frac{{(x + 5)}^{2}}{4} - y^{2} = 1$

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 $1$ .

$\frac{{(x + 5)}^{2}}{4} - \frac{y^{2}}{1} = 1$

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, $a$ .

$a = 2$

$b = 1$

$k = 0$

$h = - 5$

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

$(- 5, 0)$

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.

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

Simplify.

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

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

One to any power is one.

$\sqrt{4 + 1}$

Add $4$ and $1$ .

$\sqrt{5}$

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.

$(- 3, 0)$

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.

$(- 7, 0)$

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

$(- 3, 0), (- 7, 0)$

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.

$(- 5 + \sqrt{5}, 0)$

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.

$(- 5 - \sqrt{5}, 0)$

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

$(- 5 + \sqrt{5}, 0), (- 5 - \sqrt{5}, 0)$

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.

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

Simplify the numerator.

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

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

One to any power is one.

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

Add $4$ and $1$ .

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

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.

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

Simplify.

One to any power is one.

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

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

$y = \pm \frac{1}{2} \cdot (x - (- 5)) + 0$

Simplify to find the first asymptote.

Remove parentheses.

$y = \frac{1}{2} \cdot (x - (- 5)) + 0$

Simplify $\frac{1}{2} \cdot (x - (- 5)) + 0$ .

Simplify the expression.

Add $\frac{1}{2} \cdot (x - (- 5))$ and $0$ .

$y = \frac{1}{2} \cdot (x - (- 5))$

Multiply $- 1$ by $- 5$ .

$y = \frac{1}{2} \cdot (x + 5)$

Apply the distributive property.

$y = \frac{1}{2} x + \frac{1}{2} \cdot 5$

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

$y = \frac{x}{2} + \frac{1}{2} \cdot 5$

Combine $\frac{1}{2}$ and $5$ .

$y = \frac{x}{2} + \frac{5}{2}$

Simplify to find the second asymptote.

Remove parentheses.

$y = - \frac{1}{2} \cdot (x - (- 5)) + 0$

Simplify $- \frac{1}{2} \cdot (x - (- 5)) + 0$ .

Simplify the expression.

Add $- \frac{1}{2} \cdot (x - (- 5))$ and $0$ .

$y = - \frac{1}{2} \cdot (x - (- 5))$

Multiply $- 1$ by $- 5$ .

$y = - \frac{1}{2} \cdot (x + 5)$

Apply the distributive property.

$y = - \frac{1}{2} x - \frac{1}{2} \cdot 5$

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

$y = - \frac{x}{2} - \frac{1}{2} \cdot 5$

Multiply $- \frac{1}{2} \cdot 5$ .

Multiply $5$ by $- 1$ .

$y = - \frac{x}{2} - 5 (\frac{1}{2})$

Combine $- 5$ and $\frac{1}{2}$ .

$y = - \frac{x}{2} + \frac{- 5}{2}$

Move the negative in front of the fraction.

$y = - \frac{x}{2} - \frac{5}{2}$

This hyperbola has two asymptotes.

$y = \frac{x}{2} + \frac{5}{2}, y = - \frac{x}{2} - \frac{5}{2}$

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

Center: $(- 5, 0)$

Vertices: $(- 3, 0), (- 7, 0)$

Foci: $(- 5 + \sqrt{5}, 0), (- 5 - \sqrt{5}, 0)$

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

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

Asymptotes: $y = \frac{x}{2} + \frac{5}{2}$ , $y = - \frac{x}{2} - \frac{5}{2}$

Graph ((x+5)^2)/4-y^2=1

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