Find the Directrix (y-1)^2=-20(x-3)

${(y - 1)}^{2} = - 20 (x - 3)$

Step 1

Rewrite the equation in vertex form.

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

Rewrite the equation as $- 20 (x - 3) = {(y - 1)}^{2}$ .

$- 20 (x - 3) = {(y - 1)}^{2}$

Divide each term in $- 20 (x - 3) = {(y - 1)}^{2}$ by $- 20$ and simplify.

Divide each term in $- 20 (x - 3) = {(y - 1)}^{2}$ by $- 20$ .

$\frac{- 20 (x - 3)}{- 20} = \frac{{(y - 1)}^{2}}{- 20}$

Simplify the left side.

Cancel the common factor of $- 20$ .

Cancel the common factor.

$\frac{- 20 (x - 3)}{- 20} = \frac{{(y - 1)}^{2}}{- 20}$

Divide $x - 3$ by $1$ .

$x - 3 = \frac{{(y - 1)}^{2}}{- 20}$

Simplify the right side.

Move the negative in front of the fraction.

$x - 3 = - \frac{{(y - 1)}^{2}}{20}$

Add $3$ to both sides of the equation.

$x = - \frac{{(y - 1)}^{2}}{20} + 3$

Reorder terms.

$x = - \frac{1}{20} \cdot {(y - 1)}^{2} + 3$

Complete the square for $- \frac{1}{20} \cdot {(y - 1)}^{2} + 3$ .

Simplify the expression.

Simplify each term.

Rewrite ${(y - 1)}^{2}$ as $(y - 1) (y - 1)$ .

$- \frac{1}{20} \cdot ((y - 1) (y - 1)) + 3$

Expand $(y - 1) (y - 1)$ using the FOIL Method.

Apply the distributive property.

$- \frac{1}{20} \cdot (y (y - 1) - 1 (y - 1)) + 3$

Apply the distributive property.

$- \frac{1}{20} \cdot (y \cdot y + y \cdot - 1 - 1 (y - 1)) + 3$

Apply the distributive property.

$- \frac{1}{20} \cdot (y \cdot y + y \cdot - 1 - 1 y - 1 \cdot - 1) + 3$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$- \frac{1}{20} \cdot (y^{2} + y \cdot - 1 - 1 y - 1 \cdot - 1) + 3$

Move $- 1$ to the left of $y$ .

$- \frac{1}{20} \cdot (y^{2} - 1 \cdot y - 1 y - 1 \cdot - 1) + 3$

Rewrite $- 1 y$ as $- y$ .

$- \frac{1}{20} \cdot (y^{2} - y - 1 y - 1 \cdot - 1) + 3$

Rewrite $- 1 y$ as $- y$ .

$- \frac{1}{20} \cdot (y^{2} - y - y - 1 \cdot - 1) + 3$

Multiply $- 1$ by $- 1$ .

$- \frac{1}{20} \cdot (y^{2} - y - y + 1) + 3$

Subtract $y$ from $- y$ .

$- \frac{1}{20} \cdot (y^{2} - 2 y + 1) + 3$

Apply the distributive property.

$- \frac{1}{20} y^{2} - \frac{1}{20} (- 2 y) - \frac{1}{20} \cdot 1 + 3$

Simplify.

Combine $y^{2}$ and $\frac{1}{20}$ .

$- \frac{y^{2}}{20} - \frac{1}{20} (- 2 y) - \frac{1}{20} \cdot 1 + 3$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{20}$ into the numerator.

$- \frac{y^{2}}{20} + \frac{- 1}{20} (- 2 y) - \frac{1}{20} \cdot 1 + 3$

Factor $2$ out of $20$ .

$- \frac{y^{2}}{20} + \frac{- 1}{2 (10)} (- 2 y) - \frac{1}{20} \cdot 1 + 3$

Factor $2$ out of $- 2 y$ .

$- \frac{y^{2}}{20} + \frac{- 1}{2 (10)} (2 (- y)) - \frac{1}{20} \cdot 1 + 3$

Cancel the common factor.

$- \frac{y^{2}}{20} + \frac{- 1}{2 \cdot 10} (2 (- y)) - \frac{1}{20} \cdot 1 + 3$

Rewrite the expression.

$- \frac{y^{2}}{20} + \frac{- 1}{10} (- y) - \frac{1}{20} \cdot 1 + 3$

Combine $\frac{- 1}{10}$ and $y$ .

$- \frac{y^{2}}{20} - \frac{- y}{10} - \frac{1}{20} \cdot 1 + 3$

Multiply $- 1$ by $1$ .

$- \frac{y^{2}}{20} - \frac{- y}{10} - \frac{1}{20} + 3$

Simplify each term.

Move the negative in front of the fraction.

$- \frac{y^{2}}{20} - - \frac{y}{10} - \frac{1}{20} + 3$

Multiply $- - \frac{y}{10}$ .

Multiply $- 1$ by $- 1$ .

$- \frac{y^{2}}{20} + 1 \frac{y}{10} - \frac{1}{20} + 3$

Multiply $\frac{y}{10}$ by $1$ .

$- \frac{y^{2}}{20} + \frac{y}{10} - \frac{1}{20} + 3$

To write $3$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$- \frac{y^{2}}{20} + \frac{y}{10} - \frac{1}{20} + 3 \cdot \frac{20}{20}$

Combine $3$ and $\frac{20}{20}$ .

$- \frac{y^{2}}{20} + \frac{y}{10} - \frac{1}{20} + \frac{3 \cdot 20}{20}$

Combine the numerators over the common denominator.

$- \frac{y^{2}}{20} + \frac{y}{10} + \frac{- 1 + 3 \cdot 20}{20}$

Simplify the numerator.

Multiply $3$ by $20$ .

$- \frac{y^{2}}{20} + \frac{y}{10} + \frac{- 1 + 60}{20}$

Add $- 1$ and $60$ .

$- \frac{y^{2}}{20} + \frac{y}{10} + \frac{59}{20}$

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

$a = - \frac{1}{20}$

$b = \frac{1}{10}$

$c = \frac{59}{20}$

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

$d = \frac{\frac{1}{10}}{2 (- \frac{1}{20})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{10} \cdot \frac{1}{2 (- \frac{1}{20})}$

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

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

$d = \frac{1}{10} \cdot \frac{- 1 (- 1)}{2 (- \frac{1}{20})}$

Move the negative in front of the fraction.

$d = \frac{1}{10} (- \frac{1}{2 (\frac{1}{20})})$

Combine $2$ and $\frac{1}{20}$ .

$d = \frac{1}{10} (- \frac{1}{\frac{2}{20}})$

Cancel the common factor of $2$ and $20$ .

Factor $2$ out of $2$ .

$d = \frac{1}{10} (- \frac{1}{\frac{2 (1)}{20}})$

Cancel the common factors.

Factor $2$ out of $20$ .

$d = \frac{1}{10} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 10}})$

Cancel the common factor.

$d = \frac{1}{10} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 10}})$

Rewrite the expression.

$d = \frac{1}{10} (- \frac{1}{\frac{1}{10}})$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{10} (- (1 \cdot 10))$

Cancel the common factor of $10$ .

Factor $10$ out of $- (1 \cdot 10)$ .

$d = \frac{1}{10} (10 (- 1 \cdot (1)))$

Cancel the common factor.

$d = \frac{1}{10} (10 (- 1 \cdot (1)))$

Rewrite the expression.

$d = - 1 \cdot (1)$

Multiply $- 1$ by $1$ .

$d = - 1$

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

$e = \frac{59}{20} - \frac{{(\frac{1}{10})}^{2}}{4 (- \frac{1}{20})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{1}{10}$ .

$e = \frac{59}{20} - \frac{\frac{1^{2}}{10^{2}}}{4 (- \frac{1}{20})}$

One to any power is one.

$e = \frac{59}{20} - \frac{\frac{1}{10^{2}}}{4 (- \frac{1}{20})}$

Raise $10$ to the power of $2$ .

$e = \frac{59}{20} - \frac{\frac{1}{100}}{4 (- \frac{1}{20})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = \frac{59}{20} - \frac{\frac{1}{100}}{- 4 (\frac{1}{20})}$

Combine $- 4$ and $\frac{1}{20}$ .

$e = \frac{59}{20} - \frac{\frac{1}{100}}{\frac{- 4}{20}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 4$ and $20$ .

Factor $4$ out of $- 4$ .

$e = \frac{59}{20} - \frac{\frac{1}{100}}{\frac{4 (- 1)}{20}}$

Cancel the common factors.

Factor $4$ out of $20$ .

$e = \frac{59}{20} - \frac{\frac{1}{100}}{\frac{4 \cdot - 1}{4 \cdot 5}}$

Cancel the common factor.

$e = \frac{59}{20} - \frac{\frac{1}{100}}{\frac{4 \cdot - 1}{4 \cdot 5}}$

Rewrite the expression.

$e = \frac{59}{20} - \frac{\frac{1}{100}}{\frac{- 1}{5}}$

Move the negative in front of the fraction.

$e = \frac{59}{20} - \frac{\frac{1}{100}}{- \frac{1}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{59}{20} - (\frac{1}{100} (- 1 \cdot 5))$

Cancel the common factor of $5$ .

Factor $5$ out of $100$ .

$e = \frac{59}{20} - (\frac{1}{5 (20)} (- 1 \cdot 5))$

Factor $5$ out of $- 1 \cdot 5$ .

$e = \frac{59}{20} - (\frac{1}{5 (20)} (5 \cdot - 1))$

Cancel the common factor.

$e = \frac{59}{20} - (\frac{1}{5 \cdot 20} (5 \cdot - 1))$

Rewrite the expression.

$e = \frac{59}{20} - (\frac{1}{20} \cdot - 1)$

Combine $\frac{1}{20}$ and $- 1$ .

$e = \frac{59}{20} - \frac{- 1}{20}$

Move the negative in front of the fraction.

$e = \frac{59}{20} - - \frac{1}{20}$

Multiply $- - \frac{1}{20}$ .

Multiply $- 1$ by $- 1$ .

$e = \frac{59}{20} + 1 (\frac{1}{20})$

Multiply $\frac{1}{20}$ by $1$ .

$e = \frac{59}{20} + \frac{1}{20}$

Combine the numerators over the common denominator.

$e = \frac{59 + 1}{20}$

Add $59$ and $1$ .

$e = \frac{60}{20}$

Divide $60$ by $20$ .

$e = 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{20} {(y - 1)}^{2} + 3$ .

$- \frac{1}{20} {(y - 1)}^{2} + 3$

Set $x$ equal to the new right side.

$x = - \frac{1}{20} \cdot {(y - 1)}^{2} + 3$

Step 2

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

$a = - \frac{1}{20}$

$h = 3$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(3, 1)$

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.

$\frac{1}{4 \cdot (- \frac{1}{20})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{20})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{20})}$

Combine $4$ and $\frac{1}{20}$ .

$- \frac{1}{\frac{4}{20}}$

Cancel the common factor of $4$ and $20$ .

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{20}}$

Cancel the common factors.

Factor $4$ out of $20$ .

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 5}}$

Cancel the common factor.

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 5}}$

Rewrite the expression.

$- \frac{1}{\frac{1}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 5)$

Multiply $- (1 \cdot 5)$ .

Multiply $5$ by $1$ .

$- 1 \cdot 5$

Multiply $- 1$ by $5$ .

$- 5$

Step 5

Find the directrix.

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Substitute the known values of $p$ and $h$ into the formula and simplify.

$x = 8$

Step 6

Find the Directrix (y-1)^2=-20(x-3)

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