Precalculus Examples

${(y - 2)}^{2} = 8 (x + 1)$

Step 1

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

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Step 1.1

Rewrite the equation as $8 (x + 1) = {(y - 2)}^{2}$ .

$8 (x + 1) = {(y - 2)}^{2}$

Step 1.2

Divide each term in $8 (x + 1) = {(y - 2)}^{2}$ by $8$ and simplify.

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Step 1.2.1

Divide each term in $8 (x + 1) = {(y - 2)}^{2}$ by $8$ .

$\frac{8 (x + 1)}{8} = \frac{{(y - 2)}^{2}}{8}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $8$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{8 (x + 1)}{8} = \frac{{(y - 2)}^{2}}{8}$

Step 1.2.2.1.2

Divide $x + 1$ by $1$ .

$x + 1 = \frac{{(y - 2)}^{2}}{8}$

Step 1.3

Subtract $1$ from both sides of the equation.

$x = \frac{{(y - 2)}^{2}}{8} - 1$

Step 1.4

Reorder terms.

$x = \frac{1}{8} \cdot {(y - 2)}^{2} - 1$

Step 2

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

$a = \frac{1}{8}$

$h = - 1$

$k = 2$

Step 3

Find the vertex $(h, k)$ .

$(- 1, 2)$

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.

$\frac{1}{4 \cdot \frac{1}{8}}$

Step 4.3

Simplify.

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Step 4.3.1

Combine $4$ and $\frac{1}{8}$ .

$\frac{1}{\frac{4}{8}}$

Step 4.3.2

Cancel the common factor of $4$ and $8$ .

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Step 4.3.2.1

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{8}}$

Step 4.3.2.2

Cancel the common factors.

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Step 4.3.2.2.1

Factor $4$ out of $8$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 4.3.2.2.2

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 4.3.2.2.3

Rewrite the expression.

$\frac{1}{\frac{1}{2}}$

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Step 4.3.4

Multiply $2$ by $1$ .

$2$

Step 5

Find the focus.

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Step 5.1

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 5.2

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

$(1, 2)$

Step 6