Comets have elliptical orbits, but the orbits of some comets are so vast that they are indistinguishable from parabolas. Find an equation of the form $y = a x ^ { 2 } + b x + c$ for the parabola that closely describes the orbit of the comet shown in the illustration.

A) $y = - \frac { 1 } { 2 } x ^ { 2 } - x - \frac { 9 } { 2 }$
B) $y = \frac { 1 } { 2 } x ^ { 2 } - x - \frac { 9 } { 2 }$
C) $y = - \frac { 1 } { 2 } x ^ { 2 } - x + \frac { 9 } { 2 }$
D) $y = \frac { 1 } { 2 } x ^ { 2 } + x - \frac { 9 } { 2 }$
E) $y = \frac { 1 } { 2 } x ^ { 2 } + x + \frac { 9 } { 2 }$

Question

Comets have elliptical orbits, but the orbits of some comets are so vast that they are indistinguishable from parabolas. Find an equation of the form $y = a x ^ { 2 } + b x + c$ for the parabola that closely describes the orbit of the comet shown in the illustration.

A) $y = - \frac { 1 } { 2 } x ^ { 2 } - x - \frac { 9 } { 2 }$ 
B) $y = \frac { 1 } { 2 } x ^ { 2 } - x - \frac { 9 } { 2 }$ 
C) $y = - \frac { 1 } { 2 } x ^ { 2 } - x + \frac { 9 } { 2 }$ 
D) $y = \frac { 1 } { 2 } x ^ { 2 } + x - \frac { 9 } { 2 }$ 
E) $y = \frac { 1 } { 2 } x ^ { 2 } + x + \frac { 9 } { 2 }$

Quizplus · Accepted Answer

The parabolic orbit appears to open downwards and have a vertex below the origin. Therefore, options A and C can be eliminated. Looking at the shape of the orbit, it appears to be wider than it is tall, so the coefficient of x squared should be positive. Therefore, options D and E can be eliminated. Plugging in some of the points on the orbit, we can check which option matches the most points. Using option B, we get that the orbit passes through the points (-3, 0), (0, -9), and (3, 0), which closely matches the orbit shown in the illustration. Therefore, the answer is B.

Comets Have Elliptical Orbits, but the Orbits of Some Comets y=ax2+bx+cy = a x ^ { 2 } + b x + cy=ax2+bx+c

Comets Have Elliptical Orbits, but the Orbits of Some Comets $y = a x ^ { 2 } + b x + c$