Question - Finding the Equation of a Parabola with Specific Properties
Solution:
The question in the image is asking for an equation of a parabola that opens upwards, has a minimum value of 8, and has an axis of symmetry at $$x = 3$$. To find the correct equation, let's recall some properties of parabolas:1. A parabola that opens upwards will have the form $$f(x) = a(x - h)^2 + k$$, where $$a > 0$$.2. The vertex of the parabola is at the point $$(h, k)$$. This will also be the minimum point if the parabola opens upwards.3. The axis of symmetry is the vertical line $$x = h$$.Based on the given information, we need the parabola to open upwards which requires that $$a > 0$$, its vertex will be at $$(3, 8)$$, meaning $$h = 3$$ and $$k = 8$$, and so the axis of symmetry is $$x = 3$$.All the options in the image are in the vertex form of a parabola, $$f(x) = a(x - h)^2 + k$$. We need to find the one that matches our vertex $$(3, 8)$$ and opens upwards.- Option A has $$h = 3$$ and $$k = 8$$ which is correct, and since $$a = 1$$ (which is positive), this parabola opens upwards. This equation could be the correct answer.- Option B has the same $$h$$ and $$k$$, but a negative $$a$$, meaning this parabola opens downwards. This cannot be the correct answer.- Option C has the correct $$k$$, but $$h = -3$$, which gives us the wrong axis of symmetry. So it's not correct.- Option D has $$h = 3$$ and $$k = -8$$, which means the vertex is at $$(3, -8)$$ and this does not match our minimum value of 8.Therefore, the correct answer is Option A: $$f(x) = (x - 3)^2 + 8$$, because this is the only equation among the choices that correctly represents a parabola with the specified properties.
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