To determine the correct equation of the parabola that has a vertical axis, passes through the point [tex]\((-1,3)\)[/tex], and has its vertex at [tex]\((3,2)\)[/tex], we will use the given transformations and properties of parabolas.
1. Vertex Form of a Parabola:
The general form of a parabolic equation with a vertical axis and vertex at [tex]\((h, k)\)[/tex] is:
[tex]\[
(x - h)^2 = 4a(y - k).
\][/tex]
This represents the parabola opening either upwards or downwards.
2. Given Information:
- Vertex: [tex]\((3, 2)\)[/tex]
- Point: [tex]\((-1, 3)\)[/tex]
3. Written in Standard Form:
We rearrange the terms to:
[tex]\[
(x - 3)^2 = 4a(y - 2)
\][/tex]
4. Substitute the Given Point [tex]\((-1, 3)\)[/tex]:
To find [tex]\( a \)[/tex], substitute the point [tex]\((-1, 3)\)[/tex]:
[tex]\[
(-1 - 3)^2 = 4a(3 - 2)
\][/tex]
Simplifying, we get:
[tex]\[
16 = 4a \cdot 1
\][/tex]
Thus, [tex]\( a = 4 \)[/tex].
5. Equation of the Parabola:
Thus, the specific equation is:
[tex]\[
(x - 3)^2 = 16(y - 2) \quad \quad [1]
\][/tex]
Rearranging to make [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y - 2 = \frac{(x - 3)^2}{16} \implies y = \frac{(x - 3)^2}{16} + 2
\][/tex]
6. Expanding the Final Formula:
Expanding [tex]\((x-3)^2\)[/tex]:
[tex]\[
y = \frac{x^2 - 6x + 9}{16} + 2
\][/tex]
[tex]\[
y = \frac{x^2}{16} - \frac{6x}{16} + \frac{9}{16} + 2
\][/tex]
[tex]\[
y = \frac{x^2}{16} - \frac{6x}{16} + \frac{9}{16} + \frac{32}{16}
\][/tex]
[tex]\[
y = \frac{x^2}{16} - \frac{6x}{16} + \frac{41}{16}
\][/tex]
Thus, the final equation of the parabola in standard form is:
[tex]\[
y = \frac{x^2}{16} - \frac{6x}{16} + \frac{41}{16}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{y = \frac{x^2}{16} - \frac{6 x}{16} + \frac{41}{16}}
\][/tex]
