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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 11: Q.181E (page 1135)

In the following exercises, graph each equation.
$y^{2} - x^{2} - 4 y + 2 x - 6 = 0$

Short Answer

Expert verified

Equation given in the question represents a hyperbola.

Step by step solution

01

Identification of the conic

Given equation of the conic is,

$y^{2} - x^{2} - 4 y + 2 x - 6 = 0$

Since, coefficients have the different signs, it's a hyperbola.

02

Convert the equation of the hyperbola in the standard form

$y^{2} - x^{2} + 2 x - 4 y - 6 = 0$

$(y^{2} - 4 y + 4) - (x^{2} - 2 x + 1) - 3 - 6 = 0$

${(y - 2)}^{2} - {(x - 1)}^{2} = 9$

$\frac{{(y - 2)}^{2}}{3^{2}} - \frac{{(x - 1)}^{2}}{3^{2}} = 1$

03

Transverse axis and center of the hyperbola

Since, $y^{2}$ term of the equation is positive, transverse axis will be vertical.

Center of the hyperbola $鈫� (1, 2)$

04

Graph of the hyperbola will be,

$a^{2} = b^{2} = 9 a = b = 卤 3$

Draw a rectangle that passes through the points $3$ units above and below the center and $3$ units left and right to the center.

Sketch the asymptotes - lines through the diagonals of the rectangle.

05

Graph the branches and mark the vertices.

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Most popular questions from this chapter

Perform the indicated operation and simplify.

$\frac{4 x}{x + 2} 路 \frac{x^{2} + 5 x + 6}{12 x^{2}}$

In the following exercises, simplify each complex rational expression using either method.

$\frac{\frac{3}{b^{2} - 3 b - 40}}{\frac{5}{b + 5} - \frac{2}{b - 8}}$

Use the Quotient Property to simplify square roots.

(a) $\sqrt{\frac{27 p^{2} q}{108 p^{4} q^{3}}}$

(b) $\sqrt[3]{\frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}$

(c) $\sqrt[6]{\frac{2 m^{9} n^{7}}{128 m^{3} n}}$

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{4}{(x - 2)} 鈮� \frac{3}{(x + 1)}$

Multiply $\frac{3 {xy}^{2}}{8 y^{3}} . \frac{16 y^{2}}{24 x}$

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