Find the Vertex Form f(x)=2x^2+1/2x+8

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

Step 1

Write $f (x) = 2 x^{2} + \frac{1}{2} x + 8$ as an equation.

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

Step 2

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

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

Step 3

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

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

$a = 2$

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

$c = 8$

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

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $2$ by $2$ .

$d = \frac{1}{2} \cdot \frac{1}{4}$

Multiply $\frac{1}{2} \cdot \frac{1}{4}$ .

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$d = \frac{1}{2 \cdot 4}$

Multiply $2$ by $4$ .

$d = \frac{1}{8}$

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

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

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{1}{2}$ .

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

One to any power is one.

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

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

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

Multiply $4$ by $2$ .

$e = 8 - \frac{\frac{1}{4}}{8}$

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{1}{4} \cdot \frac{1}{8}$ .

Multiply $\frac{1}{4}$ by $\frac{1}{8}$ .

$e = 8 - \frac{1}{4 \cdot 8}$

Multiply $4$ by $8$ .

$e = 8 - \frac{1}{32}$

To write $8$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$e = 8 \cdot \frac{32}{32} - \frac{1}{32}$

Combine $8$ and $\frac{32}{32}$ .

$e = \frac{8 \cdot 32}{32} - \frac{1}{32}$

Combine the numerators over the common denominator.

$e = \frac{8 \cdot 32 - 1}{32}$

Simplify the numerator.

Multiply $8$ by $32$ .

$e = \frac{256 - 1}{32}$

Subtract $1$ from $256$ .

$e = \frac{255}{32}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + \frac{1}{8})}^{2} + \frac{255}{32}$ .

$2 {(x + \frac{1}{8})}^{2} + \frac{255}{32}$

Step 4

Set $y$ equal to the new right side.

$y = 2 {(x + \frac{1}{8})}^{2} + \frac{255}{32}$

Find the Vertex Form f(x)=2x^2+1/2x+8

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