Find the Vertex f(x)=-1/2x^2-2x+6

$f (x) = - \frac{1}{2} x^{2} - 2 x + 6$

Step 1

Rewrite the equation in vertex form.

Combine $x^{2}$ and $\frac{1}{2}$ .

$y = - \frac{x^{2}}{2} - 2 x + 6$

Complete the square for $- \frac{x^{2}}{2} - 2 x + 6$ .

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

$a = - \frac{1}{2}$

$b = - 2$

$c = 6$

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

$d = \frac{- 2}{2 (- \frac{1}{2})}$

Simplify the right side.

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

Factor $2$ out of $- 2$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Dividing two negative values results in a positive value.

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

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 2$

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

$e = 6 - \frac{{(- 2)}^{2}}{4 (- \frac{1}{2})}$

Simplify the right side.

Simplify each term.

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

$e = 6 - \frac{4}{4 (- \frac{1}{2})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = 6 - \frac{4}{- 4 (\frac{1}{2})}$

Combine $- 4$ and $\frac{1}{2}$ .

$e = 6 - \frac{4}{\frac{- 4}{2}}$

Divide $- 4$ by $2$ .

$e = 6 - \frac{4}{- 2}$

Divide $4$ by $- 2$ .

$e = 6 - - 2$

Multiply $- 1$ by $- 2$ .

$e = 6 + 2$

Add $6$ and $2$ .

$e = 8$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x + 2)}^{2} + 8$ .

$- \frac{1}{2} {(x + 2)}^{2} + 8$

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x + 2)}^{2} + 8$

Step 2

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

$a = - \frac{1}{2}$

$h = - 2$

$k = 8$

Step 3

Find the vertex $(h, k)$ .

$(- 2, 8)$

Step 4

Find the Vertex f(x)=-1/2x^2-2x+6

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