Question - Identifying a Parabolic Function with Specific Characteristics
Solution:
The question in the image is asking which equation represents a parabola that opens upwards, has a minimum value of 3, and has an axis of symmetry at $$ x = 3 $$.For a parabola in the form $$ f(x) = a(x-h)^2 + k $$, if $$ a > 0 $$, it opens upwards. The vertex of the parabola is at the point $$ (h, k) $$, where $$ h $$ is the x-coordinate of the vertex and gives the axis of symmetry, and $$ k $$ is the y-coordinate of the vertex, representing the minimum or maximum value of the function, depending on whether the parabola opens upwards or downwards.Given that the parabola has a minimum value of 3, $$ k = 3 $$, and the axis of symmetry is at $$ x = 3 $$, so $$ h = 3 $$. The value of $$ a $$ merely determines the "width" of the parabola but is always positive for an upward-opening parabola.Considering these factors, the correct equation must have a vertex at $$ (3, 3) $$ and have a positive $$ a $$ value. This leads us to an equation of the form $$ f(x) = a(x-3)^2 + 3 $$.Looking at the options in the image, the only equation that matches this form is:Option D: $$ f(x) = a(x-3)^2 + 3 $$The precise value of $$ a $$ is not given, but it is not necessary in this case because we are only concerned with the direction in which the parabola opens, the minimum value, and the axis of symmetry. All these are satisfied by Option D.
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