To determine the correct function representing the parabola with vertex at [tex]\((3,1)\)[/tex] and focus at [tex]\((3,5)\)[/tex], we can use the standard form of a parabola that opens upwards. The standard form for such a parabola is:
[tex]\[
(x - h)^2 = 4p(y - k)
\][/tex]
Where [tex]\((h, k)\)[/tex] is the vertex of the parabola, and [tex]\(p\)[/tex] is the distance from the vertex to the focus along the axis of symmetry.
First, identify the coordinates of the vertex and the focus:
- Vertex: [tex]\((3, 1)\)[/tex]
- Focus: [tex]\((3, 5)\)[/tex]
Next, calculate the distance [tex]\(p\)[/tex] between the vertex and the focus. Since both points have the same x-coordinate, the distance is simply the difference in their y-coordinates:
[tex]\[
p = 5 - 1 = 4
\][/tex]
Now, rewrite the equation of the parabola in its functional form. The general functional form derived from the standard vertex form is:
[tex]\[
y = k + \frac{1}{4p}(x - h)^2
\][/tex]
Here, [tex]\(h = 3\)[/tex], [tex]\(k = 1\)[/tex], and [tex]\(p = 4\)[/tex]. First calculate [tex]\(4p\)[/tex]:
[tex]\[
4p = 4 \cdot 4 = 16
\][/tex]
Substitute these values into the equation:
[tex]\[
y = 1 + \frac{1}{16}(x - 3)^2
\][/tex]
Thus, the function representing the graph of the parabola is:
[tex]\[
f(x) = \frac{1}{16}(x - 3)^2 + 1
\][/tex]
Hence, the correct answer is:
D. [tex]\(f(x) = \frac{1}{16}(x - 3)^2 + 1\)[/tex].
