Find the Vertex f(x)=1/2x^2-3x+7

$f (x) = \frac{1}{2} x^{2} - 3 x + 7$

Step 1

Rewrite the equation in vertex form.

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

$y = \frac{x^{2}}{2} - 3 x + 7$

Complete the square for $\frac{x^{2}}{2} - 3 x + 7$ .

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

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

$b = - 3$

$c = 7$

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

$d = \frac{- 3}{2 (\frac{1}{2})}$

Simplify the right side.

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

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

Divide $2$ by $2$ .

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

Divide $- 3$ by $1$ .

$d = - 3$

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

$e = 7 - \frac{{(- 3)}^{2}}{4 (\frac{1}{2})}$

Simplify the right side.

Simplify each term.

Raise $- 3$ to the power of $2$ .

$e = 7 - \frac{9}{4 (\frac{1}{2})}$

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

$e = 7 - \frac{9}{\frac{4}{2}}$

Divide $4$ by $2$ .

$e = 7 - \frac{9}{2}$

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

$e = 7 \cdot \frac{2}{2} - \frac{9}{2}$

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

$e = \frac{7 \cdot 2}{2} - \frac{9}{2}$

Combine the numerators over the common denominator.

$e = \frac{7 \cdot 2 - 9}{2}$

Simplify the numerator.

Multiply $7$ by $2$ .

$e = \frac{14 - 9}{2}$

Subtract $9$ from $14$ .

$e = \frac{5}{2}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(x - 3)}^{2} + \frac{5}{2}$ .

$\frac{1}{2} {(x - 3)}^{2} + \frac{5}{2}$

Set $y$ equal to the new right side.

$y = \frac{1}{2} \cdot {(x - 3)}^{2} + \frac{5}{2}$

Step 2

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

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

$h = 3$

$k = \frac{5}{2}$

Step 3

Find the vertex $(h, k)$ .

$(3, \frac{5}{2})$

Step 4

Find the Vertex f(x)=1/2x^2-3x+7

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