Algebra Examples

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

Step 1

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

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Step 1.1

Multiply both sides of the equation by $4$ .

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $4 (\frac{1}{4} \cdot (y + 3))$ .

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Step 1.2.1.1

Apply the distributive property.

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Combine $\frac{1}{4}$ and $3$ .

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

Step 1.2.1.4

Apply the distributive property.

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

Step 1.2.1.5

Cancel the common factor of $4$ .

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Step 1.2.1.5.1

Cancel the common factor.

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

Step 1.2.1.5.2

Rewrite the expression.

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

Step 1.2.1.6

Cancel the common factor of $4$ .

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Step 1.2.1.6.1

Cancel the common factor.

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

Step 1.2.1.6.2

Rewrite the expression.

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

Step 1.3

Subtract $3$ from both sides of the equation.

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

Step 2

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

$a = 4$

$h = 2$

$k = - 3$

Step 3

Find the vertex $(h, k)$ .

$(2, - 3)$

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.

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

Step 4.3

Multiply $4$ by $4$ .

$\frac{1}{16}$

Step 5

Find the directrix.

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Step 5.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 5.2

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

$y = - \frac{49}{16}$

Step 6