Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x−3)²−4(y+3)²=4

Verified step by step guidance

1. Rewrite the given equation in standard form by dividing both sides by 4 to isolate the right side as 1: \(\frac{(x-3)^2}{4} - \frac{(y+3)^2}{1} = 1\).
2. Identify the center of the hyperbola from the equation: the center is at \((h, k) = (3, -3)\).
3. Determine the values of \(a^2\) and \(b^2\) from the denominators: here, \(a^2 = 4\) and \(b^2 = 1\). Since the positive term is with \((x-3)^2\), the hyperbola opens left and right.
4. Find the vertices by moving \(a\) units left and right from the center along the x-axis: vertices are at \((3 \pm 2, -3)\).
5. Calculate the foci using \(c^2 = a^2 + b^2\): find \(c\), then locate the foci at \((3 \pm c, -3)\). Write the equations of the asymptotes using the formula \(y - k = \pm \frac{b}{a}(x - h)\), which becomes \(y + 3 = \pm \frac{1}{2}(x - 3)\).

Key Concepts

Standard Form of a Hyperbola

A hyperbola's equation can be written in standard form to identify its center, vertices, and orientation. For example, (x−h)^2/a^2 − (y−k)^2/b^2 = 1 represents a hyperbola centered at (h, k) opening left-right. Recognizing and rewriting the given equation into this form is essential for graphing and further analysis.

Foci and Vertices of a Hyperbola

Vertices are points where the hyperbola intersects its transverse axis, and foci are fixed points used to define the hyperbola. The distance from the center to each vertex is 'a', and to each focus is 'c', where c^2 = a^2 + b^2. Locating these points helps in accurately sketching the hyperbola.

Equations of Asymptotes

Asymptotes are lines that the hyperbola approaches but never touches, guiding its shape. For a hyperbola centered at (h, k), the asymptotes have equations y − k = ±(b/a)(x − h) or y − k = ±(a/b)(x − h), depending on orientation. Finding these lines is crucial for graphing the hyperbola's behavior at infinity.