Final answer:
The equation 4x^2 - 9y^2 = 36 represents a hyperbola, with vertices at (3, 0) and (-3, 0), foci at (\(\sqrt{13}\), 0) and (-\(\sqrt{13}\), 0), and asymptotes at lines y = (2/3)x and y = -(2/3)x. The correct answer is D.
Step-by-step explanation:
The equation 4x^2 - 9y^2 = 36 represents a hyperbola.
To graph this equation and find its features, we should first rewrite the equation in standard form. We do this by isolating y and placing the equation in the form (x^2/a^2) - (y^2/b^2) = 1:
\(\frac{{x^2}}{{(3)^2}} - \frac{{y^2}}{{(2)^2}} = 1\)
From this we can identify vertices at (3, 0) and (-3, 0), and foci, which can be found using the formula c^2 = a^2 + b^2. In this case, c^2 = 3^2 + 2^2, so c = \(\sqrt{{13}}\), meaning the foci are at (\(\sqrt{{13}}\), 0) and (-\(\sqrt{{13}}\), 0).
The asymptotes of the hyperbola can be described by the lines y = (b/a)x and y = -(b/a)x, or for this specific hyperbola, y = (2/3)x and y = -(2/3)x.