Suppose that the path of a newly discovered comet could be modeled by using one branch of the equation x squared over four minus y squared over nine equals one, where where distances are measured in astronomical units. Name the vertices of the hyperbola and then graph the hyperbola.
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The equation given is in the form of a hyperbola, with vertices located at (-2,0) and (2,0).
To graph the hyperbola, we can plot the vertices first, then find the asymptotes of the hyperbola. The asymptotes can be found by determining the slopes using the coefficients of the x and y terms in the equation.
The slope of the asymptotes can be calculated as ± b/a, where b is the coefficient of the y term and a is the coefficient of the x term.
So, the slopes of the asymptotes are ±3/2.
Taking the vertices (-2,0) and (2,0) as points on the asymptotes, we can draw the lines y=(3/2)x and y=(-3/2)x passing through these points.
Then, the hyperbola is the part of the graph that is bounded by the asymptotes.
The graph of the hyperbola with vertices at (-2,0) and (2,0) is shown below:
graph { (x^2/4) - (y^2/9) = 1 }