Final answer:
After substituting the points (1,4), (4,-5), and (5,-16) into the provided equations, none of the options a, b, c, or d correctly represents the parabola that passes through all three points. There appears to be an error as none of the options match the criteria.
Step-by-step explanation:
The question presented asks to determine the equation of a parabola that passes through three given points: (1,4), (4,-5), and (5,-16). We start by substituting the coordinates of these points into each of the provided equation options to see which equation satisfies all three points.
- For option a) y = -x^2 - 3x + 2, plugging in (1,4) yields -1 - 3 + 2 = -2, which does not equal 4.
- For option b) y = -x^2 - 3x - 2, plugging in (1,4) yields -1 - 3 - 2 = -6, which is not equal to 4.
- For option c) y = x^2 - 3x + 2, substituting (1,4) results in 1 - 3 + 2 = 0, again not equal to 4.
- For option d) y = x^2 - 3x - 2, substituting (1,4) gives us 1 - 3 - 2 = -4, which is not equal to 4 as well.
Since none of these options provide the correct values when we substitute the point (1,4), there seems to be an error. None of the options a, b, c, or d is the correct equation of the parabola that passes through all three points mentioned.