In Exercises 33–42, use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x+3)^2/25−y^2/16=1

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1. Identify the center of the hyperbola from the equation \(\frac{(x+3)^2}{25} - \frac{y^2}{16} = 1\). The center is at \((-3, 0)\).
2. Determine the vertices by using the value under the \(x\)-term, which is 25. The distance from the center to each vertex along the \(x\)-axis is \(\sqrt{25} = 5\). Thus, the vertices are at \((-3+5, 0)\) and \((-3-5, 0)\), which simplifies to \((2, 0)\) and \((-8, 0)\).
3. Find the foci by using the relationship \(c^2 = a^2 + b^2\), where \(a^2 = 25\) and \(b^2 = 16\). Calculate \(c\) and then determine the coordinates of the foci, which are \((-3+c, 0)\) and \((-3-c, 0)\).
4. Determine the equations of the asymptotes. For a hyperbola of the form \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\), the asymptotes are given by \(y - k = \pm \frac{b}{a}(x - h)\). Substitute \(h = -3\), \(k = 0\), \(a = 5\), and \(b = 4\) to get the equations of the asymptotes.
5. Graph the hyperbola by plotting the center, vertices, foci, and asymptotes. Draw the hyperbola opening left and right, approaching the asymptotes.

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, where (h, k) is the center, and 'a' and 'b' determine the distances to the vertices and co-vertices, respectively.

Asymptotes

Asymptotes are lines that the branches of a hyperbola approach but never touch. For a hyperbola in standard form, the equations of the asymptotes can be derived from the center and the values of 'a' and 'b'. They are given by the equations y = ±(b/a)(x-h) + k, which help in sketching the hyperbola accurately by indicating the direction of the branches.

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 by 'c', where c² = a² + b². The foci play a crucial role in defining the hyperbola's shape and are essential for understanding its reflective properties.