OG详解-OG4 数学2 Q18

Choice D is correct. It’s given that  f(x)= (x -10)(x +13), which can be rewritten as  f (x)= x² +3x -130 . Since the coefficient of the x² -term is positive, the graph of y = f (x) in the xy-plane opens upward and reaches its minimum value at its vertex. The x-coordinate of the vertex is the value of x such that f(x) reaches its minimum. For an equation in the form f(x)= ax² + bx + c, where a, b, and c are constants, the x-coordinate of the vertex is - b​2a​. For the equation f (x)= x² +3x -130 , a = 1, b = 3, and c =-130. It follows that the x-coordinate of the vertex is - 3​2(1)​, or - 32​​. Therefore, f(x) reaches its minimum when the value of x  is - 32​​. Alternate approach: The value of x  for the vertex of a parabola is the x-value of the midpoint between the two x-intercepts of the parabola. Since it’s given that f (x)= (x -10)(x +13), it follows that the two x-intercepts of the graph of y = f (x) in the xy-plane occur when x = 10 and x =-13, or at the points (10, 0) and (-13, 0). The midpoint between two points,  (x₁, y₁ ) and  (x₂, y₂ ), is (x₁+x₂2​, y₁+y₂2​​). Therefore, the midpoint between  (10, 0) and  (-13, 0) is  (10+(−13)2​, 0+0​2​) , or (−​32​, 0). It follows that f(x) reaches its minimum when the value of  x  is  - 3​2​ .
Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of y = f (x) in the xy-plane. Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of y = f (x) in the xy-plane. Choice C is incorrect and may result from conceptual or calculation errors.