Find the Vertex Form H(x)=1/2x^2-7x+7

$H (x) = \frac{1}{2} x^{2} - 7 x + 7$

Step 1

Write $H (x) = \frac{1}{2} x^{2} - 7 x + 7$ as an equation.

$y = \frac{1}{2} x^{2} - 7 x + 7$

Step 2

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

$y = \frac{x^{2}}{2} - 7 x + 7$

Step 3

Complete the square for $\frac{x^{2}}{2} - 7 x + 7$ .

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

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

$b = - 7$

$c = 7$

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

$d = \frac{- 7}{2 (\frac{1}{2})}$

Simplify the right side.

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

$d = \frac{- 7}{\frac{2}{2}}$

Divide $2$ by $2$ .

$d = \frac{- 7}{1}$

Divide $- 7$ by $1$ .

$d = - 7$

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

$e = 7 - \frac{{(- 7)}^{2}}{4 (\frac{1}{2})}$

Simplify the right side.

Simplify each term.

Raise $- 7$ to the power of $2$ .

$e = 7 - \frac{49}{4 (\frac{1}{2})}$

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

$e = 7 - \frac{49}{\frac{4}{2}}$

Divide $4$ by $2$ .

$e = 7 - \frac{49}{2}$

To write $7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = 7 \cdot \frac{2}{2} - \frac{49}{2}$

Combine $7$ and $\frac{2}{2}$ .

$e = \frac{7 \cdot 2}{2} - \frac{49}{2}$

Combine the numerators over the common denominator.

$e = \frac{7 \cdot 2 - 49}{2}$

Simplify the numerator.

Multiply $7$ by $2$ .

$e = \frac{14 - 49}{2}$

Subtract $49$ from $14$ .

$e = \frac{- 35}{2}$

Move the negative in front of the fraction.

$e = - \frac{35}{2}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(x - 7)}^{2} - \frac{35}{2}$ .

$\frac{1}{2} {(x - 7)}^{2} - \frac{35}{2}$

Step 4

Set $y$ equal to the new right side.

$y = \frac{1}{2} \cdot {(x - 7)}^{2} - \frac{35}{2}$

Find the Vertex Form H(x)=1/2x^2-7x+7

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