Question 10.3.3: Graphing a hyperbola centered at (h, k) Determine the foci a...
Graphing a hyperbola centered at (h, k)
Determine the foci and equations of the asymptotes, and sketch the graph of
\frac{(x-2)^{2}}{9}-\frac{(y+1)^{2}}{4}=1.
The graph that we seek is the graph of
\frac{x^{2}}{9}-\frac{y^{2}}{4}=1translated so that its center is (2, -1). Since a = 3, the vertices are three units from (2, -1) at (5, -1) and (-1, -1). Because b = 2, the fundamental rectangle passes through the vertices and points that are two units above and below (2, -1). Draw the fundamental rectangle through the vertices and through (2, 1) and (2, -3). Extend the diagonals of the fundamental rectangle for the asymptotes, and draw the hyperbola opening to the left and right, as shown in Fig. 10.41. Use c^{2}= a^{2}+b^{2} to get c=\sqrt{13}. The foci are on the transverse axis, \sqrt{13} units from the center (2, -1). So the foci are (2+\sqrt{13},-1) and (2-\sqrt{13},-1). The asymptotes have slopes \pm 2 / 3 and pass through (2, -1). Using the point-slope form for the equation of the line, we get the equations
y=\frac{2}{3} x-\frac{7}{3} and y=-\frac{2}{3} x+\frac{1}{3}
as the equations of the asymptotes.
Check by graphing
\begin{aligned} &y_{1}=-1+\sqrt{4(x-2)^{2} / 9-4} \\ &y_{2}=-1-\sqrt{4(x-2)^{2} / 9-4} \\ &y_{3}=(2 x-7) / 3, \text { and } \\ &y_{4}=(-2 x+1) / 3 \end{aligned}on a graphing calculator, as in Fig. 10.42.
