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

Verified step by step guidance

1. Identify the center of the hyperbola from the equation. The equation is given as \(\frac{(x+4)^2}{9} - \frac{(y+3)^2}{16} = 1\), so the center is at \((-4, -3)\).
2. Determine the values of \(a^2\) and \(b^2\) from the denominators. Here, \(a^2 = 9\) and \(b^2 = 16\), so \(a = 3\) and \(b = 4\). Since the \(x\)-term is positive and comes first, the hyperbola opens left and right (horizontal transverse axis).
3. Locate the vertices using the center and \(a\). The vertices are \(a\) units left and right from the center along the \(x\)-axis, so the vertices are at \((-4 - 3, -3)\) and \((-4 + 3, -3)\).
4. Find the foci using the relationship \(c^2 = a^2 + b^2\). Calculate \(c\) by \(c = \sqrt{9 + 16}\), then locate the foci \(c\) units left and right from the center along the \(x\)-axis, at \((-4 - c, -3)\) and \((-4 + c, -3)\).
5. Write the equations of the asymptotes. For a hyperbola with a horizontal transverse axis, the asymptotes are given by \(y - k = \pm \frac{b}{a}(x - h)\), where \((h, k)\) is the center. Substitute \(h = -4\), \(k = -3\), \(a = 3\), and \(b = 4\) to get the asymptote equations.

Key Concepts

Standard Form of a Hyperbola

A hyperbola's equation in standard form is written as (x-h)^2/a^2 - (y-k)^2/b^2 = 1 or (y-k)^2/a^2 - (x-h)^2/b^2 = 1, where (h, k) is the center. The sign in front of each term determines the hyperbola's orientation (horizontal or vertical). Understanding this form helps identify the center, vertices, and the relationship between a and b.

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 are points further from the center, located c units away, where c^2 = a^2 + b^2. These points are essential for graphing and understanding the hyperbola's shape and properties.

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' equations are y - k = ±(b/a)(x - h) for horizontal transverse axis, or y - k = ±(a/b)(x - h) for vertical. They help in sketching the hyperbola accurately.