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+4)^2/9−(y+3)^2/16=1

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1. Identify the center of the hyperbola from the equation \(\frac{(x+4)^2}{9} - \frac{(y+3)^2}{16} = 1\). The center is at \((-4, -3)\).
2. Determine the vertices by using the values of \(a^2\) and \(b^2\). Here, \(a^2 = 9\) and \(b^2 = 16\), so \(a = 3\) and \(b = 4\). The vertices are at \((-4 \pm 3, -3)\), which are \((-7, -3)\) and \((-1, -3)\).
3. Find the foci using the relationship \(c^2 = a^2 + b^2\). Calculate \(c\) and then determine the coordinates of the foci, which are at \((-4 \pm c, -3)\).
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 \(h = -4\), \(k = -3\), \(a = 3\), and \(b = 4\) to find the equations of the asymptotes.
5. Graph the hyperbola by plotting the center, vertices, foci, and asymptotes. Draw the hyperbola opening left and right through the vertices and 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's equation 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 a curve approaches as it heads towards infinity. For hyperbolas, there are two asymptotes that intersect at the center of the hyperbola. The equations of the asymptotes can be derived from the standard form of the hyperbola and are given by y - k = ±(b/a)(x - h), where (h, k) is the center of the hyperbola.

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 shape of the hyperbola and are used in various applications, including in the definition of the hyperbola itself.