Find the Directrix y^2=4x-4

$y^{2} = 4 x - 4$

Step 1

Rewrite the equation in vertex form.

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

Rewrite the equation as $4 x - 4 = y^{2}$ .

$4 x - 4 = y^{2}$

Add $4$ to both sides of the equation.

$4 x = y^{2} + 4$

Divide each term in $4 x = y^{2} + 4$ by $4$ and simplify.

Divide each term in $4 x = y^{2} + 4$ by $4$ .

$\frac{4 x}{4} = \frac{y^{2}}{4} + \frac{4}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{y^{2}}{4} + \frac{4}{4}$

Divide $x$ by $1$ .

$x = \frac{y^{2}}{4} + \frac{4}{4}$

Simplify the right side.

Divide $4$ by $4$ .

$x = \frac{y^{2}}{4} + 1$

Complete the square for $\frac{y^{2}}{4} + 1$ .

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

$a = \frac{1}{4}$

$b = 0$

$c = 1$

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

$d = \frac{0}{2 (\frac{1}{4})}$

Simplify the right side.

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

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (\frac{1}{4})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (\frac{1}{4})}$

Rewrite the expression.

$d = \frac{0}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 0 \cdot 4$

Multiply $0$ by $4$ .

$d = 0$

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

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

Simplify the right side.

Simplify each term.

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

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

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

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

Divide $4$ by $4$ .

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

Divide $0$ by $1$ .

$e = 1 - 0$

Multiply $- 1$ by $0$ .

$e = 1 + 0$

Add $1$ and $0$ .

$e = 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4} {(y + 0)}^{2} + 1$ .

$\frac{1}{4} {(y + 0)}^{2} + 1$

Set $x$ equal to the new right side.

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

Step 2

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

$a = \frac{1}{4}$

$h = 1$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(1, 0)$

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.

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

Simplify.

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

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

Simplify by dividing numbers.

Divide $4$ by $4$ .

$\frac{1}{1}$

Divide $1$ by $1$ .

$1$

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.

$x = 0$

Step 6

Find the Directrix y^2=4x-4

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