Algebra Examples

The vertex is $(- 2, 4)$ ; the directrix is $x = - 1$

Step 1

Since the directrix is horizontal, use the equation of a parabola that opens left or right.

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

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 $x$ coordinate of the vertex using the formula $x = \frac{x coordinate of focus + directrix}{2}$ . The $y$ coordinate will be the same as the $y$ coordinate of the focus.

$(\frac{- 2 - 1}{2}, 4)$

Step 2.2

Simplify the vertex.

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Step 2.2.1

Subtract $1$ from $- 2$ .

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

Step 2.2.2

Move the negative in front of the fraction.

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

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 $x$ coordinate of the vertex from the $x$ coordinate of the focus to find $p$ .

$p = - 2 + \frac{3}{2}$

Step 3.2

Simplify.

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Step 3.2.1

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

$p = - 2 \cdot \frac{2}{2} + \frac{3}{2}$

Step 3.2.2

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

$p = \frac{- 2 \cdot 2}{2} + \frac{3}{2}$

Step 3.2.3

Combine the numerators over the common denominator.

$p = \frac{- 2 \cdot 2 + 3}{2}$

Step 3.2.4

Simplify the numerator.

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Step 3.2.4.1

Multiply $- 2$ by $2$ .

$p = \frac{- 4 + 3}{2}$

Step 3.2.4.2

Add $- 4$ and $3$ .

$p = \frac{- 1}{2}$

Step 3.2.5

Move the negative in front of the fraction.

$p = - \frac{1}{2}$

Step 4

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

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

Step 5

Simplify.

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

Step 6