Find the Focus (x-1)^2=4(y-1)

${(x - 1)}^{2} = 4 (y - 1)$

Step 1

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

Rewrite the equation as $4 (y - 1) = {(x - 1)}^{2}$ .

$4 (y - 1) = {(x - 1)}^{2}$

Divide each term in $4 (y - 1) = {(x - 1)}^{2}$ by $4$ and simplify.

Divide each term in $4 (y - 1) = {(x - 1)}^{2}$ by $4$ .

$\frac{4 (y - 1)}{4} = \frac{{(x - 1)}^{2}}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 (y - 1)}{4} = \frac{{(x - 1)}^{2}}{4}$

Divide $y - 1$ by $1$ .

$y - 1 = \frac{{(x - 1)}^{2}}{4}$

Add $1$ to both sides of the equation.

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

Reorder terms.

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

Step 2

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

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

$h = 1$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(1, 1)$

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 y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

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

$(1, 2)$

Step 6

Find the Focus (x-1)^2=4(y-1)

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