Find the Focus y=1/8(x^2-4x-12)

$y = \frac{1}{8} (x^{2} - 4 x - 12)$

Step 1

Rewrite the equation in vertex form.

Simplify $\frac{1}{8} \cdot (x^{2} - 4 x - 12)$ .

Apply the distributive property.

$y = \frac{1}{8} x^{2} + \frac{1}{8} (- 4 x) + \frac{1}{8} \cdot - 12$

Simplify.

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

$y = \frac{x^{2}}{8} + \frac{1}{8} (- 4 x) + \frac{1}{8} \cdot - 12$

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

$y = \frac{x^{2}}{8} + \frac{1}{4 (2)} (- 4 x) + \frac{1}{8} \cdot - 12$

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

$y = \frac{x^{2}}{8} + \frac{1}{4 (2)} (4 (- x)) + \frac{1}{8} \cdot - 12$

Cancel the common factor.

$y = \frac{x^{2}}{8} + \frac{1}{4 \cdot 2} (4 (- x)) + \frac{1}{8} \cdot - 12$

Rewrite the expression.

$y = \frac{x^{2}}{8} + \frac{1}{2} (- x) + \frac{1}{8} \cdot - 12$

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

$y = \frac{x^{2}}{8} - \frac{x}{2} + \frac{1}{8} \cdot - 12$

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

$y = \frac{x^{2}}{8} - \frac{x}{2} + \frac{1}{4 (2)} \cdot - 12$

Factor $4$ out of $- 12$ .

$y = \frac{x^{2}}{8} - \frac{x}{2} + \frac{1}{4 \cdot 2} \cdot (4 \cdot - 3)$

Cancel the common factor.

$y = \frac{x^{2}}{8} - \frac{x}{2} + \frac{1}{4 \cdot 2} \cdot (4 \cdot - 3)$

Rewrite the expression.

$y = \frac{x^{2}}{8} - \frac{x}{2} + \frac{1}{2} \cdot - 3$

Combine $\frac{1}{2}$ and $- 3$ .

$y = \frac{x^{2}}{8} - \frac{x}{2} + \frac{- 3}{2}$

Move the negative in front of the fraction.

$y = \frac{x^{2}}{8} - \frac{x}{2} - \frac{3}{2}$

Complete the square for $\frac{x^{2}}{8} - \frac{x}{2} - \frac{3}{2}$ .

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

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

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

$c = - \frac{3}{2}$

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

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $4$ by $1$ .

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$d = - 1 \cdot 2$

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

$e = - \frac{3}{2} - \frac{{(- \frac{1}{2})}^{2}}{4 (\frac{1}{8})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{1}{2}$ .

$e = - \frac{3}{2} - \frac{{(- 1)}^{2} {(\frac{1}{2})}^{2}}{4 (\frac{1}{8})}$

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

$e = - \frac{3}{2} - \frac{1 {(\frac{1}{2})}^{2}}{4 (\frac{1}{8})}$

Apply the product rule to $\frac{1}{2}$ .

$e = - \frac{3}{2} - \frac{1 \frac{1^{2}}{2^{2}}}{4 (\frac{1}{8})}$

One to any power is one.

$e = - \frac{3}{2} - \frac{1 \frac{1}{2^{2}}}{4 (\frac{1}{8})}$

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

$e = - \frac{3}{2} - \frac{1 (\frac{1}{4})}{4 (\frac{1}{8})}$

Multiply $\frac{1}{4}$ by $1$ .

$e = - \frac{3}{2} - \frac{\frac{1}{4}}{4 (\frac{1}{8})}$

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

$e = - \frac{3}{2} - \frac{\frac{1}{4}}{\frac{4}{8}}$

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

Factor $4$ out of $4$ .

$e = - \frac{3}{2} - \frac{\frac{1}{4}}{\frac{4 (1)}{8}}$

Cancel the common factors.

Factor $4$ out of $8$ .

$e = - \frac{3}{2} - \frac{\frac{1}{4}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Cancel the common factor.

$e = - \frac{3}{2} - \frac{\frac{1}{4}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Rewrite the expression.

$e = - \frac{3}{2} - \frac{\frac{1}{4}}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$e = - \frac{3}{2} - (\frac{1}{2 (2)} \cdot 2)$

Cancel the common factor.

$e = - \frac{3}{2} - (\frac{1}{2 \cdot 2} \cdot 2)$

Rewrite the expression.

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $- 3$ .

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

Divide $- 4$ by $2$ .

$e = - 2$

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

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

Set $y$ equal to the new right side.

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

Step 2

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

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

$h = 2$

$k = - 2$

Step 3

Find the vertex $(h, k)$ .

$(2, - 2)$

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.

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

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.

$(2, 0)$

Step 6

Find the Focus y=1/8(x^2-4x-12)

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