Find the Focus 4y=x^2-12x+52

$4 y = x^{2} - 12 x + 52$

Step 1

Rewrite the equation in vertex form.

Divide each term in $4 y = x^{2} - 12 x + 52$ by $4$ and simplify.

Divide each term in $4 y = x^{2} - 12 x + 52$ by $4$ .

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{- 12 x}{4} + \frac{52}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{- 12 x}{4} + \frac{52}{4}$

Divide $y$ by $1$ .

$y = \frac{x^{2}}{4} + \frac{- 12 x}{4} + \frac{52}{4}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 12$ and $4$ .

Factor $4$ out of $- 12 x$ .

$y = \frac{x^{2}}{4} + \frac{4 (- 3 x)}{4} + \frac{52}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$y = \frac{x^{2}}{4} + \frac{4 (- 3 x)}{4 (1)} + \frac{52}{4}$

Cancel the common factor.

$y = \frac{x^{2}}{4} + \frac{4 (- 3 x)}{4 \cdot 1} + \frac{52}{4}$

Rewrite the expression.

$y = \frac{x^{2}}{4} + \frac{- 3 x}{1} + \frac{52}{4}$

Divide $- 3 x$ by $1$ .

$y = \frac{x^{2}}{4} - 3 x + \frac{52}{4}$

Divide $52$ by $4$ .

$y = \frac{x^{2}}{4} - 3 x + 13$

Complete the square for $\frac{x^{2}}{4} - 3 x + 13$ .

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

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

$b = - 3$

$c = 13$

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

$d = \frac{- 3}{2 (\frac{1}{4})}$

Simplify the right side.

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

$d = \frac{- 3}{\frac{2}{4}}$

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

Factor $2$ out of $2$ .

$d = \frac{- 3}{\frac{2 (1)}{4}}$

Cancel the common factors.

Factor $2$ out of $4$ .

$d = \frac{- 3}{\frac{2 \cdot 1}{2 \cdot 2}}$

Cancel the common factor.

$d = \frac{- 3}{\frac{2 \cdot 1}{2 \cdot 2}}$

Rewrite the expression.

$d = \frac{- 3}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$d = - 3 \cdot 2$

Multiply $- 3$ by $2$ .

$d = - 6$

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

$e = 13 - \frac{{(- 3)}^{2}}{4 (\frac{1}{4})}$

Simplify the right side.

Simplify each term.

Raise $- 3$ to the power of $2$ .

$e = 13 - \frac{9}{4 (\frac{1}{4})}$

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

$e = 13 - \frac{9}{\frac{4}{4}}$

Divide $4$ by $4$ .

$e = 13 - \frac{9}{1}$

Divide $9$ by $1$ .

$e = 13 - 1 \cdot 9$

Multiply $- 1$ by $9$ .

$e = 13 - 9$

Subtract $9$ from $13$ .

$e = 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4} {(x - 6)}^{2} + 4$ .

$\frac{1}{4} {(x - 6)}^{2} + 4$

Set $y$ equal to the new right side.

$y = \frac{1}{4} \cdot {(x - 6)}^{2} + 4$

Step 2

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

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

$h = 6$

$k = 4$

Step 3

Find the vertex $(h, k)$ .

$(6, 4)$

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.

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.

$(6, 5)$

Step 6

Find the Focus 4y=x^2-12x+52

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