Find the Focus x^2=-32y

$x^{2} = - 32 y$

Step 1

Rewrite the equation in vertex form.

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

Rewrite the equation as $- 32 y = x^{2}$ .

$- 32 y = x^{2}$

Divide each term in $- 32 y = x^{2}$ by $- 32$ and simplify.

Divide each term in $- 32 y = x^{2}$ by $- 32$ .

$\frac{- 32 y}{- 32} = \frac{x^{2}}{- 32}$

Simplify the left side.

Cancel the common factor of $- 32$ .

Cancel the common factor.

$\frac{- 32 y}{- 32} = \frac{x^{2}}{- 32}$

Divide $y$ by $1$ .

$y = \frac{x^{2}}{- 32}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{32}$

Complete the square for $- \frac{x^{2}}{32}$ .

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

$a = - \frac{1}{32}$

$b = 0$

$c = 0$

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

$d = \frac{0}{2 (- \frac{1}{32})}$

Simplify the right side.

Cancel the common factor of $0$ and $2$ .

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{1}{32})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (- \frac{1}{32})}$

Rewrite the expression.

$d = \frac{0}{- \frac{1}{32}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- 1 \cdot 32)$

Multiply $0 (- 1 \cdot 32)$ .

Multiply $- 1$ by $32$ .

$d = 0 \cdot - 32$

Multiply $0$ by $- 32$ .

$d = 0$

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

$e = 0 - \frac{0^{2}}{4 (- \frac{1}{32})}$

Simplify the right side.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$e = 0 - \frac{0}{4 (- \frac{1}{32})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = 0 - \frac{0}{- 4 (\frac{1}{32})}$

Combine $- 4$ and $\frac{1}{32}$ .

$e = 0 - \frac{0}{\frac{- 4}{32}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 4$ and $32$ .

Factor $4$ out of $- 4$ .

$e = 0 - \frac{0}{\frac{4 (- 1)}{32}}$

Cancel the common factors.

Factor $4$ out of $32$ .

$e = 0 - \frac{0}{\frac{4 \cdot - 1}{4 \cdot 8}}$

Cancel the common factor.

$e = 0 - \frac{0}{\frac{4 \cdot - 1}{4 \cdot 8}}$

Rewrite the expression.

$e = 0 - \frac{0}{\frac{- 1}{8}}$

Move the negative in front of the fraction.

$e = 0 - \frac{0}{- \frac{1}{8}}$

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (0 (- 1 \cdot 8))$

Multiply $- (0 (- 1 \cdot 8))$ .

Multiply $- 1$ by $8$ .

$e = 0 - (0 \cdot - 8)$

Multiply $0$ by $- 8$ .

$e = 0 - 0$

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Add $0$ and $0$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{32} x^{2}$ .

$- \frac{1}{32} x^{2}$

Set $y$ equal to the new right side.

$y = - \frac{1}{32} x^{2}$

Step 2

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

$a = - \frac{1}{32}$

$h = 0$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(0, 0)$

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.

$\frac{1}{4 \cdot (- \frac{1}{32})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{32})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{32})}$

Combine $4$ and $\frac{1}{32}$ .

$- \frac{1}{\frac{4}{32}}$

Cancel the common factor of $4$ and $32$ .

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{32}}$

Cancel the common factors.

Factor $4$ out of $32$ .

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

Cancel the common factor.

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

Rewrite the expression.

$- \frac{1}{\frac{1}{8}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 8)$

Multiply $- (1 \cdot 8)$ .

Multiply $8$ by $1$ .

$- 1 \cdot 8$

Multiply $- 1$ by $8$ .

$- 8$

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.

$(0, - 8)$

Step 6

Find the Focus x^2=-32y

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