Question

select the correct answer.
the vertex of a parabola is at the point (3,1), and its focus is at (3,5). what function does the graph represent?
a. $f(x)=\frac{1}{16}(x - 3)^2 - 1$
b. $f(x)=\frac{1}{4}(x + 3)^2 - 1$
c. $f(x)=\frac{1}{4}(x - 3)^2 - 1$
d. $f(x)=\frac{1}{16}(x - 3)^2 + 1$

Explanation:

Step1: Identify parabola orientation

Vertex $(3,1)$, focus $(3,5)$ share same $x$-value, so parabola opens vertically. The standard vertical parabola form with vertex $(h,k)$ is $f(x)=\frac{1}{4p}(x-h)^2 + k$, where $p$ is distance from vertex to focus.

Step2: Calculate $p$ value

$p = 5 - 1 = 4$

Step3: Substitute $h,k,p$ into formula

$h=3$, $k=1$, $p=4$. Substitute:
$f(x)=\frac{1}{4\times4}(x - 3)^2 + 1$
Simplify: $f(x)=\frac{1}{16}(x - 3)^2 + 1$

Answer:

D. $f(x)=\frac{1}{16}(x - 3)^2 + 1$