Find the Focus y^2-4y+2x+2=0

$y^{2} - 4 y + 2 x + 2 = 0$

Step 1

Rewrite the equation in vertex form.

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

Move all terms not containing $x$ to the right side of the equation.

Subtract $y^{2}$ from both sides of the equation.

$- 4 y + 2 x + 2 = - y^{2}$

Add $4 y$ to both sides of the equation.

$2 x + 2 = - y^{2} + 4 y$

Subtract $2$ from both sides of the equation.

$2 x = - y^{2} + 4 y - 2$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 y$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 y$ by $1$ .

$x = - \frac{y^{2}}{2} + 2 y + \frac{- 2}{2}$

Divide $- 2$ by $2$ .

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

Complete the square for $- \frac{y^{2}}{2} + 2 y - 1$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - \frac{1}{2}$

$b = 2$

$c = - 1$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{2}{2 (- \frac{1}{2})}$

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

$d = \frac{2}{2 (- \frac{1}{2})}$

Rewrite the expression.

$d = \frac{1}{- \frac{1}{2}}$

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$d = \frac{- 1 (- 1)}{- \frac{1}{2}}$

Move the negative in front of the fraction.

$d = - \frac{1}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$d = - (1 \cdot 2)$

Multiply $- (1 \cdot 2)$ .

Multiply $2$ by $1$ .

$d = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$d = - 2$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

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

Simplify the right side.

Simplify each term.

Raise $2$ to the power of $2$ .

$e = - 1 - \frac{4}{4 (- \frac{1}{2})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - 1 - \frac{4}{- 4 (\frac{1}{2})}$

Combine $- 4$ and $\frac{1}{2}$ .

$e = - 1 - \frac{4}{\frac{- 4}{2}}$

Divide $- 4$ by $2$ .

$e = - 1 - \frac{4}{- 2}$

Divide $4$ by $- 2$ .

$e = - 1 - - 2$

Multiply $- 1$ by $- 2$ .

$e = - 1 + 2$

Add $- 1$ and $2$ .

$e = 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(y - 2)}^{2} + 1$ .

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

Set $x$ equal to the new right side.

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

Step 2

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

$a = - \frac{1}{2}$

$h = 1$

$k = 2$

Step 3

Find the vertex $(h, k)$ .

$(1, 2)$

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.

$\frac{1}{4 \cdot (- \frac{1}{2})}$

Simplify.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{2})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{2})}$

Combine $4$ and $\frac{1}{2}$ .

$- \frac{1}{\frac{4}{2}}$

Divide $4$ by $2$ .

$- \frac{1}{2}$

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.

$(\frac{1}{2}, 2)$

Step 6

Find the Focus y^2-4y+2x+2=0

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