Find the Focus (y-2)^2=16(x-3)

${(y - 2)}^{2} = 16 (x - 3)$

Step 1

Rewrite the equation in vertex form.

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

Rewrite the equation as $16 (x - 3) = {(y - 2)}^{2}$ .

$16 (x - 3) = {(y - 2)}^{2}$

Divide each term in $16 (x - 3) = {(y - 2)}^{2}$ by $16$ and simplify.

Divide each term in $16 (x - 3) = {(y - 2)}^{2}$ by $16$ .

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

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

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

Divide $x - 3$ by $1$ .

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

Add $3$ to both sides of the equation.

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

Reorder terms.

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

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

Simplify the expression.

Simplify each term.

Rewrite ${(y - 2)}^{2}$ as $(y - 2) (y - 2)$ .

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

Expand $(y - 2) (y - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

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

Move $- 2$ to the left of $y$ .

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

Multiply $- 2$ by $- 2$ .

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

Subtract $2 y$ from $- 2 y$ .

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

Apply the distributive property.

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

Simplify.

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

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

Cancel the common factor of $4$ .

Factor $4$ out of $16$ .

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

Factor $4$ out of $- 4 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Cancel the common factor of $4$ .

Factor $4$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Combine $3$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $4$ .

$\frac{y^{2}}{16} - \frac{y}{4} + \frac{1 + 12}{4}$

Add $1$ and $12$ .

$\frac{y^{2}}{16} - \frac{y}{4} + \frac{13}{4}$

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

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

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

$c = \frac{13}{4}$

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

$d = \frac{- \frac{1}{4}}{2 (\frac{1}{16})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{1}{4} \cdot \frac{1}{2 (\frac{1}{16})}$

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

$d = - \frac{1}{4} \cdot \frac{1}{\frac{2}{16}}$

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

Factor $2$ out of $2$ .

$d = - \frac{1}{4} \cdot \frac{1}{\frac{2 (1)}{16}}$

Cancel the common factors.

Factor $2$ out of $16$ .

$d = - \frac{1}{4} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 8}}$

Cancel the common factor.

$d = - \frac{1}{4} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 8}}$

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{1}{4} (1 \cdot 8)$

Multiply $8$ by $1$ .

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

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{1}{4}$ into the numerator.

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

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

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

$e = \frac{13}{4} - \frac{{(- \frac{1}{4})}^{2}}{4 (\frac{1}{16})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{1}{4}$ .

$e = \frac{13}{4} - \frac{{(- 1)}^{2} {(\frac{1}{4})}^{2}}{4 (\frac{1}{16})}$

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

$e = \frac{13}{4} - \frac{1 {(\frac{1}{4})}^{2}}{4 (\frac{1}{16})}$

Apply the product rule to $\frac{1}{4}$ .

$e = \frac{13}{4} - \frac{1 \frac{1^{2}}{4^{2}}}{4 (\frac{1}{16})}$

One to any power is one.

$e = \frac{13}{4} - \frac{1 \frac{1}{4^{2}}}{4 (\frac{1}{16})}$

Raise $4$ to the power of $2$ .

$e = \frac{13}{4} - \frac{1 (\frac{1}{16})}{4 (\frac{1}{16})}$

Multiply $\frac{1}{16}$ by $1$ .

$e = \frac{13}{4} - \frac{\frac{1}{16}}{4 (\frac{1}{16})}$

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

$e = \frac{13}{4} - \frac{\frac{1}{16}}{\frac{4}{16}}$

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

Factor $4$ out of $4$ .

$e = \frac{13}{4} - \frac{\frac{1}{16}}{\frac{4 (1)}{16}}$

Cancel the common factors.

Factor $4$ out of $16$ .

$e = \frac{13}{4} - \frac{\frac{1}{16}}{\frac{4 \cdot 1}{4 \cdot 4}}$

Cancel the common factor.

$e = \frac{13}{4} - \frac{\frac{1}{16}}{\frac{4 \cdot 1}{4 \cdot 4}}$

Rewrite the expression.

$e = \frac{13}{4} - \frac{\frac{1}{16}}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{13}{4} - (\frac{1}{16} \cdot 4)$

Cancel the common factor of $4$ .

Factor $4$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

$e = \frac{13}{4} - \frac{1}{4}$

Combine the numerators over the common denominator.

$e = \frac{13 - 1}{4}$

Subtract $1$ from $13$ .

$e = \frac{12}{4}$

Divide $12$ by $4$ .

$e = 3$

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

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

Set $x$ equal to the new right side.

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

Step 2

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

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

$h = 3$

$k = 2$

Step 3

Find the vertex $(h, k)$ .

$(3, 2)$

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.

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

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.

$(7, 2)$

Step 6

Find the Focus (y-2)^2=16(x-3)

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