In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. y=±√x^2−2

Verified step by step guidance

1. <Step 1: Identify the standard form of the hyperbola equation.> The given equation is $y=\pm \sqrt{x^2-2}$. Rewrite it in the form $y^2=x^2-2$, which resembles the standard form of a hyperbola centered at the origin: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.
2. <Step 2: Determine the values of $a$ and $b$.> Compare $y^2=x^2-2$ with the standard form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. Here, $a^2=2$ and $b^2=1$, so $a=\sqrt{2}$ and $b=1$.
3. <Step 3: Find the vertices of the hyperbola.> For a hyperbola of the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, the vertices are located at $(\pm a,0)$. Thus, the vertices are $(\pm \sqrt{2},0)$.
4. <Step 4: Locate the foci of the hyperbola.> The distance to the foci $c$ is given by $c=\sqrt{a^2+b^2}$. Calculate $c=\sqrt{2+1}=\sqrt{3}$. The foci are at $(\pm \sqrt{3},0)$.
5. <Step 5: Find the equations of the asymptotes.> The asymptotes for a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are given by $y=\pm \frac{b}{a}x$. Substitute $a=\sqrt{2}$ and $b=1$ to get the equations $y=\pm \frac{1}{\sqrt{2}}x$.

Key Concepts

Hyperbola

A hyperbola is a type of conic section formed by the intersection of a plane and a double cone. It consists of two separate curves called branches, which are mirror images of each other. The standard form of a hyperbola can be expressed as (x-h)²/a² - (y-k)²/b² = 1 or (y-k)²/a² - (x-h)²/b² = 1, where (h, k) is the center, and 'a' and 'b' determine the distance to the vertices and co-vertices.

Asymptotes

Asymptotes are lines that a curve approaches as it heads towards infinity. For hyperbolas, the equations of the asymptotes can be derived from the standard form of the hyperbola. They provide a guide for the shape and direction of the hyperbola, typically expressed as y = k ± (b/a)(x - h) for horizontal hyperbolas and x = h ± (a/b)(y - k) for vertical hyperbolas.

Foci

The foci of a hyperbola are two fixed points located along the transverse axis, which is the line segment that connects the vertices. The distance from the center to each focus is denoted as 'c', where c² = a² + b². The foci play a crucial role in defining the hyperbola's shape and are used in the definition of the hyperbola itself, which states that the difference in distances from any point on the hyperbola to the two foci is constant.