Find the Directrix y=-4x^2+8x-21

$y = - 4 x^{2} + 8 x - 21$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 4 x^{2} + 8 x - 21$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 4$

$b = 8$

$c = - 21$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{8}{2 \cdot - 4}$

Simplify the right side.

Cancel the common factor of $8$ and $2$ .

Factor $2$ out of $8$ .

$d = \frac{2 \cdot 4}{2 \cdot - 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 4$ .

$d = \frac{2 \cdot 4}{2 (- 4)}$

Cancel the common factor.

$d = \frac{2 \cdot 4}{2 \cdot - 4}$

Rewrite the expression.

$d = \frac{4}{- 4}$

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

Factor $4$ out of $4$ .

$d = \frac{4 (1)}{- 4}$

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$d = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$d = - 1$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - 21 - \frac{8^{2}}{4 \cdot - 4}$

Simplify the right side.

Simplify each term.

Raise $8$ to the power of $2$ .

$e = - 21 - \frac{64}{4 \cdot - 4}$

Multiply $4$ by $- 4$ .

$e = - 21 - \frac{64}{- 16}$

Divide $64$ by $- 16$ .

$e = - 21 - - 4$

Multiply $- 1$ by $- 4$ .

$e = - 21 + 4$

Add $- 21$ and $4$ .

$e = - 17$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(x - 1)}^{2} - 17$ .

$- 4 {(x - 1)}^{2} - 17$

Set $y$ equal to the new right side.

$y = - 4 {(x - 1)}^{2} - 17$

Step 2

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

$a = - 4$

$h = 1$

$k = - 17$

Step 3

Find the vertex $(h, k)$ .

$(1, - 17)$

Step 4

Find $p$ , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

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

Simplify.

Multiply $4$ by $- 4$ .

$\frac{1}{- 16}$

Move the negative in front of the fraction.

$- \frac{1}{16}$

Step 5

Find the directrix.

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.

$y = k - p$

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

$y = - \frac{271}{16}$

Step 6

Find the Directrix y=-4x^2+8x-21

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