Find the Focus 1/16(y+3)^2=x+4

$\frac{1}{16} {(y + 3)}^{2} = x + 4$

Step 1

Rewrite the equation in vertex form.

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

Rewrite the equation as $x + 4 = \frac{1}{16} {(y + 3)}^{2}$ .

$x + 4 = \frac{1}{16} {(y + 3)}^{2}$

Simplify $\frac{1}{16} {(y + 3)}^{2}$ .

Rewrite.

$x + 4 = 0 + 0 + \frac{1}{16} {(y + 3)}^{2}$

Simplify by adding zeros.

$x + 4 = \frac{1}{16} {(y + 3)}^{2}$

Combine $\frac{1}{16}$ and ${(y + 3)}^{2}$ .

$x + 4 = \frac{{(y + 3)}^{2}}{16}$

Subtract $4$ from both sides of the equation.

$x = \frac{{(y + 3)}^{2}}{16} - 4$

Reorder terms.

$x = \frac{1}{16} \cdot {(y + 3)}^{2} - 4$

Complete the square for $\frac{1}{16} \cdot {(y + 3)}^{2} - 4$ .

Simplify the expression.

Simplify each term.

Rewrite ${(y + 3)}^{2}$ as $(y + 3) (y + 3)$ .

$\frac{1}{16} \cdot ((y + 3) (y + 3)) - 4$

Expand $(y + 3) (y + 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{1}{16} \cdot (y (y + 3) + 3 (y + 3)) - 4$

Apply the distributive property.

$\frac{1}{16} \cdot (y \cdot y + y \cdot 3 + 3 (y + 3)) - 4$

Apply the distributive property.

$\frac{1}{16} \cdot (y \cdot y + y \cdot 3 + 3 y + 3 \cdot 3) - 4$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$\frac{1}{16} \cdot (y^{2} + y \cdot 3 + 3 y + 3 \cdot 3) - 4$

Move $3$ to the left of $y$ .

$\frac{1}{16} \cdot (y^{2} + 3 \cdot y + 3 y + 3 \cdot 3) - 4$

Multiply $3$ by $3$ .

$\frac{1}{16} \cdot (y^{2} + 3 y + 3 y + 9) - 4$

Add $3 y$ and $3 y$ .

$\frac{1}{16} \cdot (y^{2} + 6 y + 9) - 4$

Apply the distributive property.

$\frac{1}{16} y^{2} + \frac{1}{16} (6 y) + \frac{1}{16} \cdot 9 - 4$

Simplify.

Combine $\frac{1}{16}$ and $y^{2}$ .

$\frac{y^{2}}{16} + \frac{1}{16} (6 y) + \frac{1}{16} \cdot 9 - 4$

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

$\frac{y^{2}}{16} + \frac{1}{2 (8)} (6 y) + \frac{1}{16} \cdot 9 - 4$

Factor $2$ out of $6 y$ .

$\frac{y^{2}}{16} + \frac{1}{2 (8)} (2 (3 y)) + \frac{1}{16} \cdot 9 - 4$

Cancel the common factor.

$\frac{y^{2}}{16} + \frac{1}{2 \cdot 8} (2 (3 y)) + \frac{1}{16} \cdot 9 - 4$

Rewrite the expression.

$\frac{y^{2}}{16} + \frac{1}{8} (3 y) + \frac{1}{16} \cdot 9 - 4$

Combine $3$ and $\frac{1}{8}$ .

$\frac{y^{2}}{16} + \frac{3}{8} y + \frac{1}{16} \cdot 9 - 4$

Combine $\frac{3}{8}$ and $y$ .

$\frac{y^{2}}{16} + \frac{3 y}{8} + \frac{1}{16} \cdot 9 - 4$

Combine $\frac{1}{16}$ and $9$ .

$\frac{y^{2}}{16} + \frac{3 y}{8} + \frac{9}{16} - 4$

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{y^{2}}{16} + \frac{3 y}{8} + \frac{9}{16} - 4 \cdot \frac{16}{16}$

Combine $- 4$ and $\frac{16}{16}$ .

$\frac{y^{2}}{16} + \frac{3 y}{8} + \frac{9}{16} + \frac{- 4 \cdot 16}{16}$

Combine the numerators over the common denominator.

$\frac{y^{2}}{16} + \frac{3 y}{8} + \frac{9 - 4 \cdot 16}{16}$

Simplify the numerator.

Multiply $- 4$ by $16$ .

$\frac{y^{2}}{16} + \frac{3 y}{8} + \frac{9 - 64}{16}$

Subtract $64$ from $9$ .

$\frac{y^{2}}{16} + \frac{3 y}{8} + \frac{- 55}{16}$

Move the negative in front of the fraction.

$\frac{y^{2}}{16} + \frac{3 y}{8} - \frac{55}{16}$

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

$a = \frac{1}{16}$

$b = \frac{3}{8}$

$c = - \frac{55}{16}$

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

$d = \frac{\frac{3}{8}}{2 (\frac{1}{16})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{8} \cdot \frac{1}{2 (\frac{1}{16})}$

Combine $2$ and $\frac{1}{16}$ .

$d = \frac{3}{8} \cdot \frac{1}{\frac{2}{16}}$

Cancel the common factor of $2$ and $16$ .

Factor $2$ out of $2$ .

$d = \frac{3}{8} \cdot \frac{1}{\frac{2 (1)}{16}}$

Cancel the common factors.

Factor $2$ out of $16$ .

$d = \frac{3}{8} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 8}}$

Cancel the common factor.

$d = \frac{3}{8} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 8}}$

Rewrite the expression.

$d = \frac{3}{8} \cdot \frac{1}{\frac{1}{8}}$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{8} (1 \cdot 8)$

Cancel the common factor of $8$ .

Factor $8$ out of $1 \cdot 8$ .

$d = \frac{3}{8} (8 \cdot 1)$

Cancel the common factor.

$d = \frac{3}{8} (8 \cdot 1)$

Rewrite the expression.

$d = 3$

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

$e = - \frac{55}{16} - \frac{{(\frac{3}{8})}^{2}}{4 (\frac{1}{16})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{3}{8}$ .

$e = - \frac{55}{16} - \frac{\frac{3^{2}}{8^{2}}}{4 (\frac{1}{16})}$

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

$e = - \frac{55}{16} - \frac{\frac{9}{8^{2}}}{4 (\frac{1}{16})}$

Raise $8$ to the power of $2$ .

$e = - \frac{55}{16} - \frac{\frac{9}{64}}{4 (\frac{1}{16})}$

Combine $4$ and $\frac{1}{16}$ .

$e = - \frac{55}{16} - \frac{\frac{9}{64}}{\frac{4}{16}}$

Cancel the common factor of $4$ and $16$ .

Factor $4$ out of $4$ .

$e = - \frac{55}{16} - \frac{\frac{9}{64}}{\frac{4 (1)}{16}}$

Cancel the common factors.

Factor $4$ out of $16$ .

$e = - \frac{55}{16} - \frac{\frac{9}{64}}{\frac{4 \cdot 1}{4 \cdot 4}}$

Cancel the common factor.

$e = - \frac{55}{16} - \frac{\frac{9}{64}}{\frac{4 \cdot 1}{4 \cdot 4}}$

Rewrite the expression.

$e = - \frac{55}{16} - \frac{\frac{9}{64}}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{55}{16} - (\frac{9}{64} \cdot 4)$

Cancel the common factor of $4$ .

Factor $4$ out of $64$ .

$e = - \frac{55}{16} - (\frac{9}{4 (16)} \cdot 4)$

Cancel the common factor.

$e = - \frac{55}{16} - (\frac{9}{4 \cdot 16} \cdot 4)$

Rewrite the expression.

$e = - \frac{55}{16} - \frac{9}{16}$

Combine the numerators over the common denominator.

$e = \frac{- 55 - 9}{16}$

Subtract $9$ from $- 55$ .

$e = \frac{- 64}{16}$

Divide $- 64$ by $16$ .

$e = - 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{16} {(y + 3)}^{2} - 4$ .

$\frac{1}{16} {(y + 3)}^{2} - 4$

Set $x$ equal to the new right side.

$x = \frac{1}{16} \cdot {(y + 3)}^{2} - 4$

Step 2

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

$a = \frac{1}{16}$

$h = - 4$

$k = - 3$

Step 3

Find the vertex $(h, k)$ .

$(- 4, - 3)$

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.

$\frac{1}{4 \cdot \frac{1}{16}}$

Simplify.

Combine $4$ and $\frac{1}{16}$ .

$\frac{1}{\frac{4}{16}}$

Cancel the common factor of $4$ and $16$ .

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{16}}$

Cancel the common factors.

Factor $4$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 4$

Multiply $4$ by $1$ .

$4$

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.

$(0, - 3)$

Step 6

Find the Focus 1/16(y+3)^2=x+4

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