In Exercises 33–42, use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (y+2)^2/4−(x−1)^2/16=1

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1. Identify the center of the hyperbola from the given equation \( \frac{(y+2)^2}{4} - \frac{(x-1)^2}{16} = 1 \). The center is at \((1, -2)\).
2. Determine the vertices by using the values of \(a^2\) and \(b^2\). Here, \(a^2 = 4\) and \(b^2 = 16\). The vertices are at \((1, -2 \pm 2)\), which simplifies to \((1, 0)\) and \((1, -4)\).
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 \((1, -2 \pm c)\).
4. Determine the equations of the asymptotes. For a hyperbola of the form \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \), the asymptotes are given by \( y = k \pm \frac{a}{b}(x-h) \). Substitute the values to get the equations.
5. Graph the hyperbola using the center, vertices, foci, and asymptotes. Plot the center at \((1, -2)\), the vertices at \((1, 0)\) and \((1, -4)\), and draw the asymptotes and the hyperbola curves.

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 (y-k)²/a² - (x-h)²/b² = 1 for vertical hyperbolas, where (h, k) is the center, and a and b determine the distances to the vertices and asymptotes.

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 crucial for sketching the graph accurately, as they guide the direction and shape 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 significant role in defining the hyperbola's shape and are essential for understanding its geometric properties, including how the branches relate to these points.