Find the Directrix y-2=1/4(x-5)^2

$y - 2 = \frac{1}{4} {(x - 5)}^{2}$

Step 1

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

Simplify $\frac{1}{4} {(x - 5)}^{2}$ .

Rewrite.

$y - 2 = 0 + 0 + \frac{1}{4} {(x - 5)}^{2}$

Simplify by adding zeros.

$y - 2 = \frac{1}{4} {(x - 5)}^{2}$

Combine $\frac{1}{4}$ and ${(x - 5)}^{2}$ .

$y - 2 = \frac{{(x - 5)}^{2}}{4}$

Add $2$ to both sides of the equation.

$y = \frac{{(x - 5)}^{2}}{4} + 2$

Reorder terms.

$y = \frac{1}{4} \cdot {(x - 5)}^{2} + 2$

Step 2

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

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

$h = 5$

$k = 2$

Step 3

Find the vertex $(h, k)$ .

$(5, 2)$

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.

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

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 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.

$y = 1$

Step 6