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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 10: Q. 36 (page 668)

In Problems 29鈥�36, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility. $2 x^{2} - y^{2} = 4$

Short Answer

Expert verified

$Center : (0, 0) Transverse axis : x - a x i s Vertices : (卤 \sqrt{2}, 0) Foci : (卤 \sqrt{6}, 0) Asymptotes : y = 卤 \sqrt{2} x$

Graph of the equation is:

Step by step solution

01

Step 1. Given information

Equation of hyperbola, $2 x^{2} - y^{2} = 4$

Writing the equation in standard form,

$\frac{2 x^{2}}{4} - \frac{y^{2}}{4} = 1 鈬� \frac{x^{2}}{2} - \frac{y^{2}}{4} = 1$

02

Step 2. Finding the center, transverse axis, vertices, foci, and asymptotes of hyperbola

Comparing the given equation with standard form, $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1, b^{2} = c^{2} - a^{2}$

We get,

$a = \sqrt{2}, b = 2, c = 卤 \sqrt{a^{2} + b^{2}} 鈬� c = 卤 \sqrt{2 + 2^{2}} 鈬� c = 卤 \sqrt{2 + 4} 鈬� c = 卤 \sqrt{6}$

role="math" localid="1647143938271" $Center : (0, 0) Transverse axis : x - a x i s Vertices : (卤 a, 0) = (卤 \sqrt{2}, 0) Foci : (卤 c, 0) = (卤 \sqrt{6}, 0) Asymptotes : y = 卤 \frac{b}{a} x 鈬� y = 卤 \frac{2}{\sqrt{2}} x 鈬� y = 卤 \sqrt{2} x$

03

Step 3. Graph of the equation

Graph of the equation is:

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Most popular questions from this chapter

Except for degenerate cases, the equation $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ defines $a (n)__________$ if $B^{2} - 4 A C = 0$ .

The distance $d$ from $P_{1} = (2, - 5)$ to $P_{2} = (4, - 2)$ is $d =________$ .

To graph $y = {(x + 1)}^{2} - 4$ , shift the graph of $y = x^{2}$ to the (left/right) $_____$ unit(s) and then (up/down) $_____$ unit(s).

2. To complete the square of $x^{2} - 4 x$ ,add _______.

The point that is symmetric with respect to the y-axis to the point $(- 2, 5)$ is $_____$ .

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