Question - Identifying a Parabola with Specific Characteristics
Solution:
The question in the image asks which equation represents a parabola that opens upward, has a minimum value of 3, and has an axis of symmetry at $$ x = -3 $$.Parabolas that open upward have a positive coefficient in front of the $$ x^2 $$ term. A minimum value is represented by the vertex of the parabola, and in the vertex form of a parabola, $$ y = a(x - h)^2 + k $$, where the vertex is at the point $$ (h, k) $$, $$ k $$ will be the minimum value when the parabola opens upward. The axis of symmetry is at $$ x = h $$.Looking at the options given:A. $$ f(x) = (x - (-3))^2 + 3 $$B. $$ f(x) = -(x - (-3))^2 + 6 $$C. $$ f(x) = (x - 3)^2 + 6 $$D. $$ f(x) = (x - 3)^2 + 3 $$Option A, $$ f(x) = (x - (-3))^2 + 3 $$, simplifies to $$ f(x) = (x + 3)^2 + 3 $$, which has the correct axis of symmetry at $$ x = -3 $$ and a minimum value of 3. The coefficient in front of $$ (x + 3)^2 $$ is positive, indicating that the parabola opens upward.Options B, C, and D either do not have the correct axis of symmetry, or they have a negative leading coefficient (which would mean the parabola opens downward), or they don't have the correct minimum value.Therefore, the correct option is A. $$ f(x) = (x + 3)^2 + 3 $$.
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