Find the Focus 8y=x^2-8x-16

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

Step 1

Rewrite the equation in vertex form.

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

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

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

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

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

Factor $8$ out of $- 8 x$ .

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

Cancel the common factors.

Factor $8$ out of $8$ .

$y = \frac{x^{2}}{8} + \frac{8 (- x)}{8 (1)} + \frac{- 16}{8}$

Cancel the common factor.

$y = \frac{x^{2}}{8} + \frac{8 (- x)}{8 \cdot 1} + \frac{- 16}{8}$

Rewrite the expression.

$y = \frac{x^{2}}{8} + \frac{- x}{1} + \frac{- 16}{8}$

Divide $- x$ by $1$ .

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

Divide $- 16$ by $8$ .

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

Complete the square for $\frac{x^{2}}{8} - x - 2$ .

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

$a = \frac{1}{8}$

$b = - 1$

$c = - 2$

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

$d = \frac{- 1}{2 (\frac{1}{8})}$

Simplify the right side.

Combine $2$ and $\frac{1}{8}$ .

$d = \frac{- 1}{\frac{2}{8}}$

Cancel the common factor of $2$ and $8$ .

Factor $2$ out of $2$ .

$d = \frac{- 1}{\frac{2 (1)}{8}}$

Cancel the common factors.

Factor $2$ out of $8$ .

$d = \frac{- 1}{\frac{2 \cdot 1}{2 \cdot 4}}$

Cancel the common factor.

$d = \frac{- 1}{\frac{2 \cdot 1}{2 \cdot 4}}$

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 4$

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

$e = - 2 - \frac{{(- 1)}^{2}}{4 (\frac{1}{8})}$

Simplify the right side.

Simplify each term.

Raise $- 1$ to the power of $2$ .

$e = - 2 - \frac{1}{4 (\frac{1}{8})}$

Combine $4$ and $\frac{1}{8}$ .

$e = - 2 - \frac{1}{\frac{4}{8}}$

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

Factor $4$ out of $4$ .

$e = - 2 - \frac{1}{\frac{4 (1)}{8}}$

Cancel the common factors.

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$e = - 2 - (1 \cdot 2)$

Multiply $- (1 \cdot 2)$ .

Multiply $2$ by $1$ .

$e = - 2 - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$e = - 2 - 2$

Subtract $2$ from $- 2$ .

$e = - 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{8} {(x - 4)}^{2} - 4$ .

$\frac{1}{8} {(x - 4)}^{2} - 4$

Set $y$ equal to the new right side.

$y = \frac{1}{8} \cdot {(x - 4)}^{2} - 4$

Step 2

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

$a = \frac{1}{8}$

$h = 4$

$k = - 4$

Step 3

Find the vertex $(h, k)$ .

$(4, - 4)$

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.

$\frac{1}{4 \cdot \frac{1}{8}}$

Simplify.

Combine $4$ and $\frac{1}{8}$ .

$\frac{1}{\frac{4}{8}}$

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

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{8}}$

Cancel the common factors.

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Multiply $2$ by $1$ .

$2$

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.

$(4, - 2)$

Step 6

Find the Focus 8y=x^2-8x-16

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