Question 11.4.5: Graphing a Hyperbola Centered at (0, 0) OBJECTIVE Sketch the...
Graphing a Hyperbola Centered at (0, 0)
OBJECTIVE
Sketch the graph of a hyperbola in the form
(i) \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 or (ii) \frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1.
Step 1 Write the equation in standard form. Determine transverse axis and the orientation of the hyperbola.
Step 2 Locate vertices and the endpoints of the conjugate axis.
Step 3 Lightly sketch the fundamental rectangle by drawing dashed lines parallel to the coordinate axes through the points in Step 2.
Step 4 Sketch the asymptotes. Extend the diagonals of the fundamental rectangle. These are the asymptotes.
Step 5 Sketch the graph. Draw both branches of the hyperbola through the vertices, approaching the asymptotes. See Figures 25 and 26.The foci are located on the transverse axis, c units from the center, where c² = a² +b².
Sketch the graph of
a. 16x² – 9y² = 144. b. 25y² – 4x² = 100.
Minus sign precedes y²-term, and transverse axis is along the x-axis; hyperbola opens left and right.
Because a^{2}=9, a=3 . Also, b^{2}=16 ; \text { so } b=4 . The vertices are (±3, 0), and the endpoints of the conjugate axis are (0, ±4).
Draw dashed lines x = 3, x = -3, y = 4,and y = -4 to form the fundamental rectangle with vertices (3, 4), (-3, 4), (-3, -4), and (3, -4).
\begin{aligned}25 y^{2}-4 x^{2} &=100 \\\frac{25 y^{2}}{100}-\frac{4 x^{2}}{100} &=\frac{100}{100} \\\frac{y^{2}}{4}-\frac{x^{2}}{25} &=1\end{aligned}Minus sign precedes x²-term, and transverse axis is along the y-axis; hyperbola opens up and down.
Because a^{2}=4, a=2. Also, b^{2}=25 ; \text { so } b=5. The vertices are (0, ±2) and the endpoints of the conjugate axis are (±5, 0).
Draw dashed lines y = 2, y = -2, x = 5, and x = -5 to form the fundamental rectangle with vertices (5, 2), (-5, 2), (-5, -2), and (5, -2).
