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

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

Step 1

Rewrite the equation in vertex form.

Add $3$ to both sides of the equation.

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

Complete the square for ${(2 - x)}^{2} + 3$ .

Simplify the expression.

Simplify each term.

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

$(2 - x) (2 - x) + 3$

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

Apply the distributive property.

$2 (2 - x) - x (2 - x) + 3$

Apply the distributive property.

$2 \cdot 2 + 2 (- x) - x (2 - x) + 3$

Apply the distributive property.

$2 \cdot 2 + 2 (- x) - x \cdot 2 - x (- x) + 3$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$4 + 2 (- x) - x \cdot 2 - x (- x) + 3$

Multiply $- 1$ by $2$ .

$4 - 2 x - x \cdot 2 - x (- x) + 3$

Multiply $2$ by $- 1$ .

$4 - 2 x - 2 x - x (- x) + 3$

Rewrite using the commutative property of multiplication.

$4 - 2 x - 2 x - 1 \cdot - 1 x \cdot x + 3$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$4 - 2 x - 2 x - 1 \cdot - 1 (x \cdot x) + 3$

Multiply $x$ by $x$ .

$4 - 2 x - 2 x - 1 \cdot - 1 x^{2} + 3$

Multiply $- 1$ by $- 1$ .

$4 - 2 x - 2 x + 1 x^{2} + 3$

Multiply $x^{2}$ by $1$ .

$4 - 2 x - 2 x + x^{2} + 3$

Subtract $2 x$ from $- 2 x$ .

$4 - 4 x + x^{2} + 3$

Add $4$ and $3$ .

$- 4 x + x^{2} + 7$

Reorder $- 4 x$ and $x^{2}$ .

$x^{2} - 4 x + 7$

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

$a = 1$

$b = - 4$

$c = 7$

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

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

Cancel the common factor of $- 4$ and $2$ .

Factor $2$ out of $- 4$ .

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

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 2$ by $1$ .

$d = - 2$

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

$e = 7 - \frac{{(- 4)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

Rewrite $- 4$ as $- 1 (4)$ .

$e = 7 - \frac{{(- 1 (4))}^{2}}{4 \cdot 1}$

Apply the product rule to $- 1 (4)$ .

$e = 7 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot 1}$

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

$e = 7 - \frac{1 \cdot 4^{2}}{4 \cdot 1}$

Multiply $4^{2}$ by $1$ .

$e = 7 - \frac{4^{2}}{4 \cdot 1}$

Factor $4$ out of $4^{2}$ .

$e = 7 - \frac{4 \cdot 4}{4 \cdot 1}$

Cancel the common factors.

Factor $4$ out of $4 \cdot 1$ .

$e = 7 - \frac{4 \cdot 4}{4 (1)}$

Cancel the common factor.

$e = 7 - \frac{4 \cdot 4}{4 \cdot 1}$

Rewrite the expression.

$e = 7 - \frac{4}{1}$

Divide $4$ by $1$ .

$e = 7 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 7 - 4$

Subtract $4$ from $7$ .

$e = 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 2)}^{2} + 3$ .

${(x - 2)}^{2} + 3$

Set $y$ equal to the new right side.

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

Step 2

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

$a = 1$

$h = 2$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(2, 3)$

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.

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

Cancel the common factor of $1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{4}$

Step 5

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

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

$y = \frac{11}{4}$

Step 6

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

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