Find the Vertex f(x)=-x^2-2x+9

$f (x) = - x^{2} - 2 x + 9$

Step 1

Rewrite the equation in vertex form.

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

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

$a = - 1$

$b = - 2$

$c = 9$

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

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

Simplify the right side.

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

Factor $2$ out of $- 2$ .

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

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

$d = - 1 \cdot - 1$

Rewrite $- 1 \cdot - 1$ as $- - 1$ .

$d = - - 1$

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

$e = 9 - \frac{{(- 2)}^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

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

$e = 9 - \frac{4}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = 9 - \frac{4}{- 4}$

Divide $4$ by $- 4$ .

$e = 9 - - 1$

Multiply $- 1$ by $- 1$ .

$e = 9 + 1$

Add $9$ and $1$ .

$e = 10$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + 1)}^{2} + 10$ .

$- {(x + 1)}^{2} + 10$

Set $y$ equal to the new right side.

$y = - {(x + 1)}^{2} + 10$

Step 2

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

$a = - 1$

$h = - 1$

$k = 10$

Step 3

Find the vertex $(h, k)$ .

$(- 1, 10)$

Step 4

Find the Vertex f(x)=-x^2-2x+9

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