Algebra Examples

$y - 14 = - {(x - 3)}^{2}$

Step 1

Add $14$ to both sides of the equation.

$y = - {(x - 3)}^{2} + 14$

Step 2

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

$a = - 1$

$h = 3$

$k = 14$

Step 3

Find the vertex $(h, k)$ .

$(3, 14)$

Step 4

Find $p$ , the distance from the vertex to the focus.

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Step 4.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 4.2

Substitute the value of $a$ into the formula.

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 5

Find the directrix.

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Step 5.1

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.

$y = k - p$

Step 5.2

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

$y = \frac{57}{4}$

Step 6