Graph the hyperbola. Locate the foci and find the equations of the asymptotes. ((x-2)^2)/25 - ((y+3)^2)/16 = 1

Verified step by step guidance

1. Identify the standard form of the hyperbola equation: $\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1$. Here, $h=2$, $k=-3$, $a^2=25$, and $b^2=16$.
2. Determine the center of the hyperbola, which is at the point $(h,k)=(2,-3)$.
3. Calculate the foci using the formula $c^2=a^2+b^2$. Find $c$ and then the foci at $(h\pm c,k)$.
4. Find the equations of the asymptotes. For a hyperbola of this form, the asymptotes are given by $y=k\pm \frac{b}{a}(x-h)$. Substitute the known values to find the equations.
5. Sketch the hyperbola using the center, vertices, foci, and asymptotes to guide the shape of the graph.

Key Concepts

Hyperbola Definition

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 for horizontal hyperbolas, where (h, k) is the center, and a and b determine the distances to the vertices and co-vertices.

Foci of a Hyperbola

The foci of a hyperbola are two fixed points located along the transverse axis, which is the line segment that connects the vertices of the hyperbola. The distance from the center to each focus is given by the formula c = √(a² + b²), where a and b are the distances to the vertices and co-vertices, respectively. The foci play a crucial role in defining the shape and properties of the hyperbola.

Asymptotes of a Hyperbola

Asymptotes are lines that the branches of a hyperbola approach but never touch. For a hyperbola in the standard form (x-h)²/a² - (y-k)²/b² = 1, the equations of the asymptotes can be derived as y - k = ±(b/a)(x - h). These lines provide a visual guide to the behavior of the hyperbola as it extends towards infinity, indicating the direction in which the branches open.