Algebra Examples

(2, 0)

$(2, 0)$

x = - 2

$x = - 2$

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)

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

Step 2

Find the vertex.

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Step 2.1

The vertex

(h, k)

$(h, k)$ is halfway between the directrix and focus. Find the

x

$x$ coordinate of the vertex using the formula

x = \frac{x coordinate of focus + directrix}{2}

$x = \frac{x coordinate of focus + directrix}{2}$ . The

y

$y$ coordinate will be the same as the

y

$y$ coordinate of the focus.

(\frac{2 - 2}{2}, 0)

$(\frac{2 - 2}{2}, 0)$

Step 2.2

Simplify the vertex.

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Step 2.2.1

Cancel the common factor of

2 - 2

$2 - 2$ and

2

$2$ .

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Step 2.2.1.1

Factor

2

$2$ out of

2

$2$ .

(\frac{2 \cdot 1 - 2}{2}, 0)

$(\frac{2 \cdot 1 - 2}{2}, 0)$

Step 2.2.1.2

Factor

2

$2$ out of

- 2

$- 2$ .

(\frac{2 \cdot 1 + 2 \cdot - 1}{2}, 0)

$(\frac{2 \cdot 1 + 2 \cdot - 1}{2}, 0)$

Step 2.2.1.3

Factor

2

$2$ out of

2 \cdot 1 + 2 \cdot - 1

$2 \cdot 1 + 2 \cdot - 1$ .

(\frac{2 \cdot (1 - 1)}{2}, 0)

$(\frac{2 \cdot (1 - 1)}{2}, 0)$

Step 2.2.1.4

Cancel the common factors.

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Step 2.2.1.4.1

Factor

2

$2$ out of

2

$2$ .

(\frac{2 \cdot (1 - 1)}{2 (1)}, 0)

$(\frac{2 \cdot (1 - 1)}{2 (1)}, 0)$

Step 2.2.1.4.2

Cancel the common factor.

(\frac{2 \cdot (1 - 1)}{2 \cdot 1}, 0)

$(\frac{2 \cdot (1 - 1)}{2 \cdot 1}, 0)$

Step 2.2.1.4.3

Rewrite the expression.

(\frac{1 - 1}{1}, 0)

$(\frac{1 - 1}{1}, 0)$

Step 2.2.1.4.4

Divide

1 - 1

$1 - 1$ by

1

$1$ .

(1 - 1, 0)

$(1 - 1, 0)$

(1 - 1, 0)

$(1 - 1, 0)$

(1 - 1, 0)

$(1 - 1, 0)$

Step 2.2.2

Subtract

1

$1$ from

1

$1$ .

(0, 0)

$(0, 0)$

(0, 0)

$(0, 0)$

(0, 0)

$(0, 0)$

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 |

$| p |$ . Subtract the

x

$x$ coordinate of the vertex from the

x

$x$ coordinate of the focus to find

p

$p$ .

p = 2 - 0

$p = 2 - 0$

Step 3.2

Subtract

0

$0$ from

2

$2$ .

p = 2

$p = 2$

p = 2

$p = 2$

Step 4

Substitute in the known values for the variables into the equation

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

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

{(y - 0)}^{2} = 4 (2) (x - 0)

${(y - 0)}^{2} = 4 (2) (x - 0)$

Step 5

Simplify.

y^{2} = 8 x

$y^{2} = 8 x$

Step 6