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

Verified step by step guidance

1. Identify the center of the hyperbola from the equation \(\frac{(y+2)^2}{4} - \frac{(x-1)^2}{16} = 1\). The center is at the point \((h, k) = (1, -2)\).
2. Determine the values of \(a^2\) and \(b^2\) from the denominators. Here, \(a^2 = 4\) and \(b^2 = 16\). Since the \(y\)-term is positive and comes first, the hyperbola opens vertically.
3. Find the vertices by moving \(a\) units up and down from the center along the \(y\)-axis. The vertices are at \((1, -2 \pm a)\), where \(a = \sqrt{4}\).
4. Calculate the foci using the relationship \(c^2 = a^2 + b^2\). Find \(c\) and then locate the foci at \((1, -2 \pm c)\) along the vertical axis.
5. Write the equations of the asymptotes using the formula for vertical hyperbolas: \(y = k \pm \frac{a}{b}(x - h)\). Substitute \(a\), \(b\), \(h\), and \(k\) to get the asymptote equations.

Key Concepts

Standard Form of a Hyperbola

A hyperbola's equation in standard form is either (y-k)^2/a^2 - (x-h)^2/b^2 = 1 or (x-h)^2/a^2 - (y-k)^2/b^2 = 1, where (h, k) is the center. The sign and position of terms determine the hyperbola's orientation (vertical or horizontal). This form helps identify key features like vertices, foci, and asymptotes.

Vertices and Foci of a Hyperbola

Vertices are points on the hyperbola closest to the center, located a units away along the transverse axis. Foci lie further out, at a distance c from the center, where c^2 = a^2 + b^2. Knowing vertices and foci is essential for graphing and understanding the hyperbola's shape.

Equations of Asymptotes

Asymptotes are lines that the hyperbola approaches but never touches, passing through the center. For a vertical hyperbola, asymptotes have equations y - k = ±(a/b)(x - h). These lines guide the hyperbola's shape and are crucial for accurate graphing.