Find the Vertex y=-3x^2-12x-9

$y = - 3 x^{2} - 12 x - 9$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 3 x^{2} - 12 x - 9$ .

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

$a = - 3$

$b = - 12$

$c = - 9$

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

$d = \frac{- 12}{2 \cdot - 3}$

Simplify the right side.

Cancel the common factor of $- 12$ and $2$ .

Factor $2$ out of $- 12$ .

$d = \frac{2 \cdot - 6}{2 \cdot - 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 3$ .

$d = \frac{2 \cdot - 6}{2 (- 3)}$

Cancel the common factor.

$d = \frac{2 \cdot - 6}{2 \cdot - 3}$

Rewrite the expression.

$d = \frac{- 6}{- 3}$

Cancel the common factor of $- 6$ and $- 3$ .

Factor $- 3$ out of $- 6$ .

$d = \frac{- 3 (2)}{- 3}$

Cancel the common factors.

Factor $- 3$ out of $- 3$ .

$d = \frac{- 3 \cdot 2}{- 3 \cdot 1}$

Cancel the common factor.

$d = \frac{- 3 \cdot 2}{- 3 \cdot 1}$

Rewrite the expression.

$d = \frac{2}{1}$

Divide $2$ by $1$ .

$d = 2$

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

$e = - 9 - \frac{{(- 12)}^{2}}{4 \cdot - 3}$

Simplify the right side.

Simplify each term.

Raise $- 12$ to the power of $2$ .

$e = - 9 - \frac{144}{4 \cdot - 3}$

Multiply $4$ by $- 3$ .

$e = - 9 - \frac{144}{- 12}$

Divide $144$ by $- 12$ .

$e = - 9 - - 12$

Multiply $- 1$ by $- 12$ .

$e = - 9 + 12$

Add $- 9$ and $12$ .

$e = 3$

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

$- 3 {(x + 2)}^{2} + 3$

Set $y$ equal to the new right side.

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

Step 2

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

$a = - 3$

$h = - 2$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(- 2, 3)$

Step 4

Find the Vertex y=-3x^2-12x-9

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