Math  /  Algebra

QuestionFind the center, the vertices, the foci, and the asymptotes. Then draw the graph. (x2)29(y+5)216=1\frac{(x-2)^{2}}{9}-\frac{(y+5)^{2}}{16}=1

Studdy Solution
Sketch the graph of the hyperbola using the center, vertices, foci, and asymptotes.
1. Plot the center at (2,5)(2, -5).
2. Plot the vertices at (5,5)(5, -5) and (1,5)(-1, -5).
3. Plot the foci at (7,5)(7, -5) and (3,5)(-3, -5).
4. Draw the asymptotes using the equations y=5+43(x2)y = -5 + \frac{4}{3}(x-2) and y=543(x2)y = -5 - \frac{4}{3}(x-2).
5. Sketch the hyperbola opening left and right, approaching the asymptotes as it extends.

The center is (2,5)(2, -5), the vertices are (5,5)(5, -5) and (1,5)(-1, -5), the foci are (7,5)(7, -5) and (3,5)(-3, -5), and the asymptotes are y=5+43(x2)y = -5 + \frac{4}{3}(x-2) and y=543(x2)y = -5 - \frac{4}{3}(x-2).

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