Find the Focus x=12(y-4)^2+3

$x = 12 {(y - 4)}^{2} + 3$

Step 1

Use the vertex form, $x = a {(y - k)}^{2} + h$ , to determine the values of $a$ , $h$ , and $k$ .

$a = 12$

$h = 3$

$k = 4$

Step 2

Find the vertex $(h, k)$ .

$(3, 4)$

Step 3

Find $p$ , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot 12}$

Multiply $4$ by $12$ .

$\frac{1}{48}$

Step 4

Find the focus.

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(\frac{145}{48}, 4)$

Step 5