Algebra Examples

$y + 7 = {(x - 4)}^{2}$

Step 1

Subtract $7$ from both sides of the equation.

$y = {(x - 4)}^{2} - 7$

Step 2

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

$a = 1$

$h = 4$

$k = - 7$

Step 3

Find the vertex $(h, k)$ .

$(4, - 7)$

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.

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

Step 4.3

Cancel the common factor of $1$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

$\frac{1}{4}$

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.

$y = - \frac{29}{4}$

Step 6