Graph ((x-4)^2)/25-((y-1)^2)/9=1

$\frac{{(x - 4)}^{2}}{25} - \frac{{(y - 1)}^{2}}{9} = 1$

Step 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 $1$ .

$\frac{{(x - 4)}^{2}}{25} - \frac{{(y - 1)}^{2}}{9} = 1$

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, $a$ .

$a = 5$

$b = 3$

$k = 1$

$h = 4$

Step 4

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

$(4, 1)$

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.

$\sqrt{{(5)}^{2} + {(3)}^{2}}$

Simplify.

Raise $5$ to the power of $2$ .

$\sqrt{25 + {(3)}^{2}}$

Raise $3$ to the power of $2$ .

$\sqrt{25 + 9}$

Add $25$ and $9$ .

$\sqrt{34}$

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.

$(9, 1)$

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.

$(- 1, 1)$

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

$(9, 1), (- 1, 1)$

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.

$(4 + \sqrt{34}, 1)$

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.

$(4 - \sqrt{34}, 1)$

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

$(4 + \sqrt{34}, 1), (4 - \sqrt{34}, 1)$

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.

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

Simplify the numerator.

Raise $5$ to the power of $2$ .

$\frac{\sqrt{25 + 3^{2}}}{5}$

Raise $3$ to the power of $2$ .

$\frac{\sqrt{25 + 9}}{5}$

Add $25$ and $9$ .

$\frac{\sqrt{34}}{5}$

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.

$\frac{3^{2}}{\sqrt{34}}$

Simplify.

Raise $3$ to the power of $2$ .

$\frac{9}{\sqrt{34}}$

Multiply $\frac{9}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\frac{9}{\sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}}$

Combine and simplify the denominator.

Multiply $\frac{9}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\frac{9 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Raise $\sqrt{34}$ to the power of $1$ .

$\frac{9 \sqrt{34}}{{\sqrt{34}}^{1} \sqrt{34}}$

Raise $\sqrt{34}$ to the power of $1$ .

$\frac{9 \sqrt{34}}{{\sqrt{34}}^{1} {\sqrt{34}}^{1}}$

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

$\frac{9 \sqrt{34}}{{\sqrt{34}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{9 \sqrt{34}}{{\sqrt{34}}^{2}}$

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{9 \sqrt{34}}{{(34^{\frac{1}{2}})}^{2}}$

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

$\frac{9 \sqrt{34}}{34^{\frac{1}{2} \cdot 2}}$

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

$\frac{9 \sqrt{34}}{34^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{9 \sqrt{34}}{34^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{9 \sqrt{34}}{34^{1}}$

Evaluate the exponent.

$\frac{9 \sqrt{34}}{34}$

Step 10

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

$y = \pm \frac{3}{5} \cdot (x - (4)) + 1$

Step 11

Simplify to find the first asymptote.

Remove parentheses.

$y = \frac{3}{5} \cdot (x - (4)) + 1$

Simplify $\frac{3}{5} \cdot (x - (4)) + 1$ .

Simplify each term.

Multiply $- 1$ by $4$ .

$y = \frac{3}{5} \cdot (x - 4) + 1$

Apply the distributive property.

$y = \frac{3}{5} x + \frac{3}{5} \cdot - 4 + 1$

Combine $\frac{3}{5}$ and $x$ .

$y = \frac{3 x}{5} + \frac{3}{5} \cdot - 4 + 1$

Multiply $\frac{3}{5} \cdot - 4$ .

Combine $\frac{3}{5}$ and $- 4$ .

$y = \frac{3 x}{5} + \frac{3 \cdot - 4}{5} + 1$

Multiply $3$ by $- 4$ .

$y = \frac{3 x}{5} + \frac{- 12}{5} + 1$

Move the negative in front of the fraction.

$y = \frac{3 x}{5} - \frac{12}{5} + 1$

Simplify the expression.

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

$y = \frac{3 x}{5} - \frac{12}{5} + \frac{5}{5}$

Combine the numerators over the common denominator.

$y = \frac{3 x}{5} + \frac{- 12 + 5}{5}$

Add $- 12$ and $5$ .

$y = \frac{3 x}{5} + \frac{- 7}{5}$

Move the negative in front of the fraction.

$y = \frac{3 x}{5} - \frac{7}{5}$

Step 12

Simplify to find the second asymptote.

Remove parentheses.

$y = - \frac{3}{5} \cdot (x - (4)) + 1$

Simplify $- \frac{3}{5} \cdot (x - (4)) + 1$ .

Simplify each term.

Multiply $- 1$ by $4$ .

$y = - \frac{3}{5} \cdot (x - 4) + 1$

Apply the distributive property.

$y = - \frac{3}{5} x - \frac{3}{5} \cdot - 4 + 1$

Combine $x$ and $\frac{3}{5}$ .

$y = - \frac{x \cdot 3}{5} - \frac{3}{5} \cdot - 4 + 1$

Multiply $- \frac{3}{5} \cdot - 4$ .

Multiply $- 4$ by $- 1$ .

$y = - \frac{x \cdot 3}{5} + 4 (\frac{3}{5}) + 1$

Combine $4$ and $\frac{3}{5}$ .

$y = - \frac{x \cdot 3}{5} + \frac{4 \cdot 3}{5} + 1$

Multiply $4$ by $3$ .

$y = - \frac{x \cdot 3}{5} + \frac{12}{5} + 1$

Move $3$ to the left of $x$ .

$y = - \frac{3 x}{5} + \frac{12}{5} + 1$

Simplify the expression.

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

$y = - \frac{3 x}{5} + \frac{12}{5} + \frac{5}{5}$

Combine the numerators over the common denominator.

$y = - \frac{3 x}{5} + \frac{12 + 5}{5}$

Add $12$ and $5$ .

$y = - \frac{3 x}{5} + \frac{17}{5}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{3 x}{5} - \frac{7}{5}, y = - \frac{3 x}{5} + \frac{17}{5}$

Step 14

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

Center: $(4, 1)$

Vertices: $(9, 1), (- 1, 1)$

Foci: $(4 + \sqrt{34}, 1), (4 - \sqrt{34}, 1)$

Eccentricity: $\frac{\sqrt{34}}{5}$

Focal Parameter: $\frac{9 \sqrt{34}}{34}$

Asymptotes: $y = \frac{3 x}{5} - \frac{7}{5}$ , $y = - \frac{3 x}{5} + \frac{17}{5}$

Step 15

Graph ((x-4)^2)/25-((y-1)^2)/9=1

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