Finding the Equation of a Parabola with Specific Properties
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.
