Precalculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Divide each term in $4 (y - 4) = {(x + 2)}^{2}$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 (y - 4) = {(x + 2)}^{2}$ by $4$ .

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide $y - 4$ by $1$ .

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

Step 1.3

Add $4$ to both sides of the equation.

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

Step 1.4

Reorder terms.

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

Step 2

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

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

$h = - 2$

$k = 4$

Step 3

Find the vertex $(h, k)$ .

$(- 2, 4)$

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

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

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

Step 4.3.2

Simplify by dividing numbers.

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Step 4.3.2.1

Divide $4$ by $4$ .

$\frac{1}{1}$

Step 4.3.2.2

Divide $1$ 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 y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 5.2

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

$(- 2, 5)$

Step 6