Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $y = \pm \sqrt{x^{2} - 2}$

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Rewrite the given equation to identify the form of the hyperbola. The equation is $y = \pm \sqrt{x^2 - 2}$. Square both sides to eliminate the square root, giving $y^2 = x^2 - 2$.

Rearrange the equation to standard form by moving all terms to one side: $y^2 - x^2 = -2$. Multiply both sides by $-1$ to get $x^2 - y^2 = 2$.

Recognize that the equation $\frac{x^2}{2} - \frac{y^2}{2} = 1$ is the standard form of a hyperbola centered at the origin with a horizontal transverse axis, where $a^2 = 2$ and $b^2 = 2$.

Find the vertices by using $a = \sqrt{2}$. Since the transverse axis is horizontal, the vertices are at $(\pm a, 0)$, which are $(\pm \sqrt{2}, 0)$.

Find the equations of the asymptotes using the formula $y = \pm \frac{b}{a} x$. Since $a = \sqrt{2}$ and $b = \sqrt{2}$, the asymptotes are $y = \pm x$. To find the foci, calculate $c$ where $c^2 = a^2 + b^2$, so $c = \sqrt{2 + 2} = \sqrt{4} = 2$, and the foci are at $(\pm c, 0)$ or $(\pm 2, 0)$.

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Hyperbola Definition and Properties

A hyperbola is a type of conic section formed by the difference of distances to two fixed points called foci. It consists of two separate branches and has key features such as vertices, foci, and asymptotes. Understanding its standard form and geometric properties is essential for graphing and analyzing hyperbolas.

Vertices and Foci of a Hyperbola

Vertices are the points where each branch of the hyperbola intersects its principal axis, and foci are fixed points inside the branches that define the curve. The distance between the center and vertices is 'a', and the distance to foci is 'c', related by the equation c² = a² + b². Locating these points helps in accurately sketching the hyperbola.

Equations and Graphing of Asymptotes

Asymptotes are straight lines that the hyperbola approaches but never touches. For hyperbolas centered at the origin, asymptotes have equations derived from the relationship between 'a' and 'b', typically y = ±(b/a)x or y = ±(a/b)x. Identifying asymptotes guides the shape and orientation of the hyperbola on the graph.

Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes.y=±x2−2y = \(\pm\]\sqrt{x^2 - 2}\)