Find the Vertex y=1/2x^2+2x-8

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

Rewrite the equation in vertex form.

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

$y = \frac{x^{2}}{2} + 2 x - 8$

Complete the square for $\frac{x^{2}}{2} + 2 x - 8$ .

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

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

$b = 2$

$c = - 8$

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 $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

$e = - 8 - \frac{2^{2}}{4 (\frac{1}{2})}$

Simplify the right side.

Simplify each term.

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

$e = - 8 - \frac{4}{4 (\frac{1}{2})}$

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

$e = - 8 - \frac{4}{\frac{4}{2}}$

Divide $4$ by $2$ .

$e = - 8 - \frac{4}{2}$

Divide $4$ by $2$ .

$e = - 8 - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$e = - 8 - 2$

Subtract $2$ from $- 8$ .

$e = - 10$

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

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

Set $y$ equal to the new right side.

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

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

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

$h = - 2$

$k = - 10$

Find the vertex $(h, k)$ .

$(- 2, - 10)$

Find the Vertex y=1/2x^2+2x-8

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