OG详解-OG4 数学1 Q26

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 = a(x -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 = ax² -18ax + 81a -14. Therefore, the equation of the parabola, y = ax² -18ax + 81a -14, can be written in the form y = ax² + 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 =264​​, which is a positive value. Therefore, if the equation of the parabola is written in the form y = ax² + bx +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 = ax² + bx +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 = ax² + bx +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 = ax² + bx +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.