Algebra Examples

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

Step 1

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

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Step 1.1

Rewrite the equation as $x - 1 = - \frac{1}{4} {(y + 2)}^{2}$ .

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

Step 1.2

Simplify $- \frac{1}{4} {(y + 2)}^{2}$ .

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Step 1.2.1

Rewrite.

$x - 1 = 0 + 0 - \frac{1}{4} {(y + 2)}^{2}$

Step 1.2.2

Simplify by adding zeros.

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

Step 1.2.3

Combine ${(y + 2)}^{2}$ and $\frac{1}{4}$ .

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

Step 1.3

Add $1$ to both sides of the equation.

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

Step 1.4

Reorder terms.

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

Step 2

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

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

$h = 1$

$k = - 2$

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.

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

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 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.

$(0, - 2)$

Step 6