Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x+3)²/25−y²/16=1

Verified step by step guidance

1. Identify the center of the hyperbola from the equation \(\frac{(x+3)^2}{25} - \frac{y^2}{16} = 1\). The center is at \((-3, 0)\) because the equation is in the form \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) where \((h, k)\) is the center.
2. Determine the values of \(a^2\) and \(b^2\) from the denominators: \(a^2 = 25\) and \(b^2 = 16\). Then find \(a\) and \(b\) by taking the square roots: \(a = 5\) and \(b = 4\).
3. Locate the vertices of the hyperbola. Since the \(x\)-term is positive and comes first, the transverse axis is horizontal. The vertices are \(a\) units left and right from the center along the \(x\)-axis, so the vertices are at \((-3 - 5, 0)\) and \((-3 + 5, 0)\).
4. Find the foci using the relationship \(c^2 = a^2 + b^2\). Calculate \(c\) by taking the square root of \(a^2 + b^2\). The foci are located \(c\) units left and right from the center along the \(x\)-axis, so their coordinates are \((-3 - c, 0)\) and \((-3 + c, 0)\).
5. Write the equations of the asymptotes. For a hyperbola with a horizontal transverse axis, the asymptotes are given by \(y - k = \pm \frac{b}{a}(x - h)\). Substitute \(h = -3\), \(k = 0\), \(a = 5\), and \(b = 4\) to get the equations of the asymptotes.

Key Concepts

Standard Form of a Hyperbola

A hyperbola's equation in standard form is either (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 given equation (x+3)^2/25 - y^2/16 = 1 shows a horizontal transverse axis centered at (-3,0), with a^2 = 25 and b^2 = 16.

Vertices and Foci of a Hyperbola

Vertices lie a units from the center along the transverse axis, so here at (-3 ± 5, 0). Foci are located c units from the center, where c^2 = a^2 + b^2. For this hyperbola, c = √(25 + 16) = √41, so foci are at (-3 ± √41, 0).

Equations of the Asymptotes

Asymptotes of a hyperbola with horizontal transverse axis have equations y - k = ±(b/a)(x - h). For this hyperbola, the asymptotes are y = ±(4/5)(x + 3), which guide the shape of the hyperbola and are shown in the graph.