In the xy -plane, a parabola has vertex ( 9 , - 14 ) and intersects the x -axis at

Question

In the xy -plane, a parabola has vertex ( 9 , - 14 ) and intersects the x -axis at two points. If the equation of the parabola is written in the form y = a x 2 + b x + c , where a , b , and c are constants, which of the following could be the value of a + b + c ?

Accepted Answer

Choice D is correct. The equation of a parabola in the xy -plane can be written in the form y = a x - h 2 + k , where a is a constant and h , k is the vertex of the parabola. If a is positive, the parabola will open upward, and if a is negative, the parabola will open downward. It&rsquo;s given that the parabola has vertex 9 , - 14 . Substituting 9 for h and - 14 for k in the equation y = a x - h 2 + k gives y = a x - 9 2 - 14 , which can be rewritten as y = a x - 9 x - 9 - 14 , or y = a x 2 - 18 x + 81 - 14 . Distributing the factor of a on the right-hand side of this equation yields y = a x 2 - 18 a x + 81 a - 14 . Therefore, the equation of the parabola, y = a x 2 - 18 a x + 81 a - 14 , can be written in the form y = a x 2 + b x + c , where a = a , b = - 18 a , and c = 81 a - 14 . Substituting - 18 a for b and 81 a - 14 for c in the expression a + b + c yields a + - 18 a + 81 a - 14 , or 64 a - 14 . Since the vertex of the parabola, 9 , - 14 , is below the x -axis, and it&rsquo;s given that the parabola intersects the x -axis at two points, the parabola must open upward. Therefore, the constant a must have a positive value. Setting the expression 64 a - 14 equal to the value in choice D yields 64 a - 14 = - 12 . Adding 14 to both sides of this equation yields 64 a = 2 . Dividing both sides of this equation by 64 yields a = 2 64 , which is a positive value. Therefore, if the equation of the parabola is written in the form y = a x 2 + b x + c , where a , b , and c are constants, the value of a + b + c could be - 12 . Choice A is incorrect. If the equation of a parabola with a vertex at 9 , - 14 is written in the form y = a x 2 + b x + c , where a , b , and c are constants and a + b + c = - 23 , then the value of a will be negative, which means the parabola will open downward, not upward, and will intersect the x -axis at zero points, not two points. Choice B is incorrect. If the equation of a parabola with a vertex at 9 , - 14 is written in the form y = a x 2 + b x + c , where a , b , and c are constants and a + b + c = - 19 , then the value of a will be negative, which means the parabola will open downward, not upward, and will intersect the x -axis at zero points, not two points. Choice C is incorrect. If the equation of a parabola with a vertex at 9 , - 14 is written in the form y = a x 2 + b x + c , where a , b , and c are constants and a + b + c = - 14 , then the value of a will be 0 , which is inconsistent with the equation of a parabola.

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