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

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

Step 1

Isolate $x$ to the left side of the equation.

Divide each term in $4 (x - 2) = {(y + 3)}^{2}$ by $4$ and simplify.

Divide each term in $4 (x - 2) = {(y + 3)}^{2}$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x - 2$ by $1$ .

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

Add $2$ to both sides of the equation.

$x = \frac{{(y + 3)}^{2}}{4} + 2$

Reorder terms.

$x = \frac{1}{4} \cdot {(y + 3)}^{2} + 2$

Step 2

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

$a = \frac{1}{4}$

$h = 2$

$k = - 3$

Step 3

Find the vertex $(h, k)$ .

$(2, - 3)$

Step 4

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.

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

Simplify.

Combine $4$ and $\frac{1}{4}$ .

$\frac{1}{\frac{4}{4}}$

Simplify by dividing numbers.

Divide $4$ by $4$ .

$\frac{1}{1}$

Divide $1$ by $1$ .

$1$

Step 5

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.

$(3, - 3)$

Step 6