Question - Determining the Equation of an Upward-Opening Parabola with a Given Minimum Value and Axis of Symmetry
Solution:
The question is asking for the equation of a parabola that opens upward, has a minimum value of 3, and an axis of symmetry at x=3.A parabola that opens upward must have a positive coefficient before the squared term in its equation. The minimum value of the parabola would be the y-coordinate of the vertex. Because we know the axis of symmetry is x = 3, this means the x-coordinate of the vertex is 3.The general form of a parabola's equation is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. If the parabola opens upwards and has a minimum value of 3, the k value (which represents the y-coordinate of the vertex) would be 3.Given this information, we can rule out option B and option C, because they have a minus sign before the squared term, which indicates the parabola opens downward, and because their k values are -6 and therefore can't represent a minimum value of 3.The only option that fits all criteria is option A: f(x) = (x - 3)^2 + 3, as this represents a parabola with the vertex at (3, 3), which means it opens upwards and has a minimum value of 3, and the axis of symmetry is at x = 3.Therefore, the correct answer is A.
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