Precalculus Examples

${(y + 2)}^{2} = - 4 (x - 7)$

Step 1

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

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Step 1.1

Rewrite the equation as $- 4 (x - 7) = {(y + 2)}^{2}$ .

$- 4 (x - 7) = {(y + 2)}^{2}$

Step 1.2

Divide each term in $- 4 (x - 7) = {(y + 2)}^{2}$ by $- 4$ and simplify.

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Step 1.2.1

Divide each term in $- 4 (x - 7) = {(y + 2)}^{2}$ by $- 4$ .

$\frac{- 4 (x - 7)}{- 4} = \frac{{(y + 2)}^{2}}{- 4}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 4 (x - 7)}{- 4} = \frac{{(y + 2)}^{2}}{- 4}$

Step 1.2.2.1.2

Divide $x - 7$ by $1$ .

$x - 7 = \frac{{(y + 2)}^{2}}{- 4}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x - 7 = - \frac{{(y + 2)}^{2}}{4}$

Step 1.3

Add $7$ to both sides of the equation.

$x = - \frac{{(y + 2)}^{2}}{4} + 7$

Step 1.4

Reorder terms.

$x = - \frac{1}{4} \cdot {(y + 2)}^{2} + 7$

Step 2

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

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

$h = 7$

$k = - 2$

Step 3

Find the vertex $(h, k)$ .

$(7, - 2)$

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.

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

Step 4.3

Simplify.

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Step 4.3.1

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

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Step 4.3.1.1

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

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

Step 4.3.1.2

Move the negative in front of the fraction.

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

Step 4.3.2

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

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

Step 4.3.3

Divide $4$ by $4$ .

$- \frac{1}{1}$

Step 4.3.4

Cancel the common factor of $1$ .

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Step 4.3.4.1

Cancel the common factor.

$- \frac{1}{1}$

Step 4.3.4.2

Rewrite the expression.

$- 1 \cdot 1$

Step 4.3.5

Multiply $- 1$ by $1$ .

$- 1$

Step 5

Find the directrix.

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Step 5.1

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.

$x = h - p$

Step 5.2

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

$x = 8$

Step 6