Algebra Examples

$(2, - 1)$ $y = - \frac{1}{2}$

Step 1

Since the directrix is vertical, use the equation of a parabola that opens up or down.

${(x - h)}^{2} = 4 p (y - k)$

Step 2

Find the vertex.

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Step 2.1

The vertex $(h, k)$ is halfway between the directrix and focus. Find the $y$ coordinate of the vertex using the formula $y = \frac{y coordinate of focus + directrix}{2}$ . The $x$ coordinate will be the same as the $x$ coordinate of the focus.

$(2, \frac{- 1 - \frac{1}{2}}{2})$

Step 2.2

Simplify the vertex.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$(2, \frac{- 1 \cdot \frac{2}{2} - \frac{1}{2}}{2})$

Step 2.2.1.2

Combine $- 1$ and $\frac{2}{2}$ .

$(2, \frac{\frac{- 1 \cdot 2}{2} - \frac{1}{2}}{2})$

Step 2.2.1.3

Combine the numerators over the common denominator.

$(2, \frac{\frac{- 1 \cdot 2 - 1}{2}}{2})$

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Multiply $- 1$ by $2$ .

$(2, \frac{\frac{- 2 - 1}{2}}{2})$

Step 2.2.1.4.2

Subtract $1$ from $- 2$ .

$(2, \frac{\frac{- 3}{2}}{2})$

Step 2.2.1.5

Move the negative in front of the fraction.

$(2, \frac{- \frac{3}{2}}{2})$

Step 2.2.2

Multiply the numerator by the reciprocal of the denominator.

$(2, - \frac{3}{2} \cdot \frac{1}{2})$

Step 2.2.3

Multiply $- \frac{3}{2} \cdot \frac{1}{2}$ .

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Step 2.2.3.1

Multiply $\frac{1}{2}$ by $\frac{3}{2}$ .

$(2, - \frac{3}{2 \cdot 2})$

Step 2.2.3.2

Multiply $2$ by $2$ .

$(2, - \frac{3}{4})$

Step 3

Find the distance from the focus to the vertex.

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Step 3.1

The distance from the focus to the vertex and from the vertex to the directrix is $| p |$ . Subtract the $y$ coordinate of the vertex from the $y$ coordinate of the focus to find $p$ .

$p = - 1 + \frac{3}{4}$

Step 3.2

Simplify.

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Step 3.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$p = - 1 \cdot \frac{4}{4} + \frac{3}{4}$

Step 3.2.2

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

$p = \frac{- 1 \cdot 4}{4} + \frac{3}{4}$

Step 3.2.3

Combine the numerators over the common denominator.

$p = \frac{- 1 \cdot 4 + 3}{4}$

Step 3.2.4

Simplify the numerator.

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Step 3.2.4.1

Multiply $- 1$ by $4$ .

$p = \frac{- 4 + 3}{4}$

Step 3.2.4.2

Add $- 4$ and $3$ .

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

Step 3.2.5

Move the negative in front of the fraction.

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

Step 4

Substitute in the known values for the variables into the equation ${(x - h)}^{2} = 4 p (y - k)$ .

${(x - 2)}^{2} = 4 (- \frac{1}{4}) (y + \frac{3}{4})$

Step 5

Simplify.

${(x - 2)}^{2} = - 1 (y + \frac{3}{4})$

Step 6