Find the Directrix y=-3x^2+2x-2

$y = - 3 x^{2} + 2 x - 2$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 3 x^{2} + 2 x - 2$ .

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

$a = - 3$

$b = 2$

$c = - 2$

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 $d = \frac{b}{2 a}$ .

$d = \frac{2}{2 \cdot - 3}$

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

$d = \frac{2}{2 \cdot - 3}$

Rewrite the expression.

$d = \frac{1}{- 3}$

Move the negative in front of the fraction.

$d = - \frac{1}{3}$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = - 2 - \frac{2^{2}}{4 \cdot - 3}$

Simplify the right side.

Simplify each term.

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

$e = - 2 - \frac{4}{4 \cdot - 3}$

Multiply $4$ by $- 3$ .

$e = - 2 - \frac{4}{- 12}$

Cancel the common factor of $4$ and $- 12$ .

Factor $4$ out of $4$ .

$e = - 2 - \frac{4 (1)}{- 12}$

Cancel the common factors.

Factor $4$ out of $- 12$ .

$e = - 2 - \frac{4 \cdot 1}{4 \cdot - 3}$

Cancel the common factor.

$e = - 2 - \frac{4 \cdot 1}{4 \cdot - 3}$

Rewrite the expression.

$e = - 2 - \frac{1}{- 3}$

Move the negative in front of the fraction.

$e = - 2 - - \frac{1}{3}$

Multiply $- - \frac{1}{3}$ .

Multiply $- 1$ by $- 1$ .

$e = - 2 + 1 (\frac{1}{3})$

Multiply $\frac{1}{3}$ by $1$ .

$e = - 2 + \frac{1}{3}$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$e = - 2 \cdot \frac{3}{3} + \frac{1}{3}$

Combine $- 2$ and $\frac{3}{3}$ .

$e = \frac{- 2 \cdot 3}{3} + \frac{1}{3}$

Combine the numerators over the common denominator.

$e = \frac{- 2 \cdot 3 + 1}{3}$

Simplify the numerator.

Multiply $- 2$ by $3$ .

$e = \frac{- 6 + 1}{3}$

Add $- 6$ and $1$ .

$e = \frac{- 5}{3}$

Move the negative in front of the fraction.

$e = - \frac{5}{3}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(x - \frac{1}{3})}^{2} - \frac{5}{3}$ .

$- 3 {(x - \frac{1}{3})}^{2} - \frac{5}{3}$

Set $y$ equal to the new right side.

$y = - 3 {(x - \frac{1}{3})}^{2} - \frac{5}{3}$

Step 2

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

$a = - 3$

$h = \frac{1}{3}$

$k = - \frac{5}{3}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{1}{3}, - \frac{5}{3})$

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.

$\frac{1}{4 \cdot - 3}$

Simplify.

Multiply $4$ by $- 3$ .

$\frac{1}{- 12}$

Move the negative in front of the fraction.

$- \frac{1}{12}$

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.

$y = - \frac{19}{12}$

Step 6

Find the Directrix y=-3x^2+2x-2

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