Find the Focus y^2-8y+4x+16=0

$y^{2} - 8 y + 4 x + 16 = 0$

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.

$- 8 y + 4 x + 16 = - y^{2}$

Add $8 y$ to both sides of the equation.

$4 x + 16 = - y^{2} + 8 y$

Subtract $16$ from both sides of the equation.

$4 x = - y^{2} + 8 y - 16$

Divide each term in $4 x = - y^{2} + 8 y - 16$ by $4$ and simplify.

Divide each term in $4 x = - y^{2} + 8 y - 16$ by $4$ .

$\frac{4 x}{4} = \frac{- y^{2}}{4} + \frac{8 y}{4} + \frac{- 16}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- y^{2}}{4} + \frac{8 y}{4} + \frac{- 16}{4}$

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{4} + \frac{8 y}{4} + \frac{- 16}{4}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{4} + \frac{8 y}{4} + \frac{- 16}{4}$

Cancel the common factor of $8$ and $4$ .

Factor $4$ out of $8 y$ .

$x = - \frac{y^{2}}{4} + \frac{4 (2 y)}{4} + \frac{- 16}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$x = - \frac{y^{2}}{4} + \frac{4 (2 y)}{4 (1)} + \frac{- 16}{4}$

Cancel the common factor.

$x = - \frac{y^{2}}{4} + \frac{4 (2 y)}{4 \cdot 1} + \frac{- 16}{4}$

Rewrite the expression.

$x = - \frac{y^{2}}{4} + \frac{2 y}{1} + \frac{- 16}{4}$

Divide $2 y$ by $1$ .

$x = - \frac{y^{2}}{4} + 2 y + \frac{- 16}{4}$

Divide $- 16$ by $4$ .

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

Complete the square for $- \frac{y^{2}}{4} + 2 y - 4$ .

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

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

$b = 2$

$c = - 4$

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 $d = \frac{b}{2 a}$ .

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

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$d = \frac{1}{- \frac{1}{4}}$

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

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

$d = \frac{- 1 (- 1)}{- \frac{1}{4}}$

Move the negative in front of the fraction.

$d = - \frac{1}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$d = - (1 \cdot 4)$

Multiply $- (1 \cdot 4)$ .

Multiply $4$ by $1$ .

$d = - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$d = - 4$

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 $e = c - \frac{b^{2}}{4 a}$ .

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

Simplify the right side.

Simplify each term.

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

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

Simplify the denominator.

Multiply $- 1$ by $4$ .

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

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

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

Divide $- 4$ by $4$ .

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

Divide $4$ by $- 1$ .

$e = - 4 - - 4$

Multiply $- 1$ by $- 4$ .

$e = - 4 + 4$

Add $- 4$ and $4$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{4} {(y - 4)}^{2} + 0$ .

$- \frac{1}{4} {(y - 4)}^{2} + 0$

Set $x$ equal to the new right side.

$x = - \frac{1}{4} \cdot {(y - 4)}^{2} + 0$

Step 2

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

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

$h = 0$

$k = 4$

Step 3

Find the vertex $(h, k)$ .

$(0, 4)$

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.

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

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

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

Move the negative in front of the fraction.

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

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

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

Divide $4$ by $4$ .

$- \frac{1}{1}$

Cancel the common factor of $1$ .

Cancel the common factor.

$- \frac{1}{1}$

Rewrite the expression.

$- 1 \cdot 1$

Multiply $- 1$ 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.

$(- 1, 4)$

Step 6

Find the Focus y^2-8y+4x+16=0

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