If a and b are constants, and the graph of g(x) = (x - a)2 + b has its minimum at g(6), which of the following pairs of points could also be on the graph of g(x) in the xy-plane?

Step 1: Identify the properties of the given function. The function is given as g(x) = (x-a)^2 + b. This is the vertex form of a parabola, y = (x-h)^2 + k, where (h, k) is the vertex. For g(x), the vertex is (a, b). Since the coefficient of (x-a)^2 is positive (it's 1), the parabola opens upwards, meaning the vertex is the minimum point. Step 2: Use the given information about the minimum. The problem states that the graph has its minimum at g(6). This means the x-coordinate of the vertex is 6. Therefore, a = 6. The function can now be written as g(x) = (x-6)^2 + b. Step 3: Understand the symmetry of the parabola. A parabola is symmetric about its axis of symmetry, which is a vertical line passing through its vertex. For g(x) = (x-6)^2 + b, the axis of symmetry is x=6. This means that if two distinct points on the parabola have the same y-coordinate, their x-coordinates must be equidistant from the axis of symmetry x=6. If (x_1, y) and (x_2, y) are two points on the parabola, then y = (x_1-6)^2 + b and y = (x_2-6)^2 + b. This implies (x_1-6)^2 = (x_2-6)^2. Taking the square root of both sides gives x_1-6 = ± (x_2-6). Since the points are distinct, x_1 ≠ x_2, so we must have x_1-6 = -(x_2-6). x_1-6 = -x_2+6 x_1+x_2 = 12. So, for any two distinct points with the same y-coordinate on the graph, their x-coordinates must sum to 12. Step 4: Check the given options. We need to find a pair of points that could be on the graph. A) (5, -9) and (8, -9) The y-coordinates are the same (-9). Check the sum of the x-coordinates: 5 + 8 = 13. Since 13 ≠ 12, this pair of points cannot be on the graph. B) (0, 6) and (10, 10) The y-coordinates are different. Let's see if both points can satisfy g(x) = (x-6)^2 + b for some value of b. For (0, 6): 6 = (0-6)^2 + b 6 = (-6)^2 + b 6 = 36 + b b = -30. So, g(x) = (x-6)^2 - 30. Now check (10, 10) with this function: 10 = (10-6)^2 - 30 10 = (4)^2 - 30 10 = 16 - 30 10 = -14. This is false, so this pair of points cannot be on the graph. C) (5, 6) and (7, 8) The y-coordinates are different. For (5, 6): 6 = (5-6)^2 + b 6 = (-1)^2 + b 6 = 1 + b b = 5. So, g(x) = (x-6)^2 + 5. Now check (7, 8) with this function: 8 = (7-6)^2 + 5 8 = (1)^2 + 5 8 = 1 + 5 8 = 6. This is false, so this pair of points cannot be on the graph. D) (2, -5) and (10, -5) The y-coordinates are the same (-5). Check the sum of the x-coordinates: 2 + 10 = 12. Since 12 = 12, this pair of points satisfies the symmetry condition. Let's verify by finding b. For (2, -5): -5 = (2-6)^2 + b -5 = (-4)^2 + b -5 = 16 + b b = -21. So, g(x) = (x-6)^2 - 21. Now check (10, -5) with this function: -5 = (10-6)^2 - 21 -5 = (4)^2 - 21 -5 = 16 - 21 -5 = -5. This is true. Both points lie on the graph when b = -21. The final answer is D. 3 done, 2 left today. You're making progress.