In the xy-plane, a parabola has vertex (9, −14) and intersects the x-axis at two points. If the equation ofthe parabola is written in the formy = ax2 + bx + c, where a, b, and c are constants, which of the following could be the value of a + b + c ?

Explanation

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'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= (ax - 9) (x - 9) - 14, or y = a(x² - 18x + 81) - 14. Distributing the factor of a on the right-hand side of this equation yields y = ax - 18ax + 81a - 14. Therefore, the equation of the parabola, y = ax - 18ax + 81a - 14, can be written in the form y = ax+bx+c, where a = a, b= -18a, and c = 81a - 14. Substituting -18a for b and 81a - 14 for c in the expression a+b+c yields (a)+(-18a)+(81a - 14), or 64a - 14. Since the vertex of the parabola, (9, -14), is below the x-axis, and it'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. 64a - 14 equal to the value in choice D yields 64a - 14 = -12. Adding 14 to both sides of this equation yields 64a = 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 = ax + bx + c, where a, b, and c are constants, the value of a + b + c could be -12.