Answer
Refer to the graph below.
Work Step by Step
RECALL:
The graph of the equation $x^2-y^2=r^2$ is a hyperbola whose center is at the origin and whose vertices are $r$ units to the left and to the right of its center.
The lines $y=x$ and $y=-x$ serve as the asymptotes of the curve.
The given equation can be written as $x^2-y^2=6^2$.
Thus, its graph is a hyperbola with center at the origin and whose vertices is $6$ units to the left and to the right of its center.
The lines $y=x$ and $y=-x$ serve as asymptotes of the hyperbola.
To graph this equation, perform the following steps:
(1) Plot the center $(0, 0)$.
(2) Plot the following points:
first vertex: $(-6, 0)$
second vertex: $(6, 0)$
(3) Sketch the graph of $y=x$ and $y=-x$ using broken lines as they are not part of the hyperbola but only serve as asymptotes.
(4) Sketch a parabola that opens:
(a) to the left, passing through $(-6, 0)$ and asymptotic the lines $y=x$ below and $y=-x$ above; and
(b) to the right, passing through $(6, 0)$ and asymptotic the lines $y=x$ above and $y=-x$ below.
Refer to the graph below for the graph with the asymptotes, and the graph in the answer part above for the graph without the asymptotes.
