Find the Standard Form of the Parabola f(x)=2x^2-8x+14

$f (x) = 2 x^{2} - 8 x + 14$

Step 1

Write $f (x) = 2 x^{2} - 8 x + 14$ as an equation.

$y = 2 x^{2} - 8 x + 14$

Step 2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 x^{2} - 8 x + 14 = y$

Step 3

Subtract $14$ from both sides of the equation.

$2 x^{2} - 8 x = y - 14$

Step 4

Complete the square.

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

$a = 2$

$b = - 8$

$c = 0 = y - 14$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 14$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 8}{2 \cdot 2}$

Simplify the right side.

Cancel the common factor of $- 8$ and $2$ .

Factor $2$ out of $- 8$ .

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

Cancel the common factors.

Factor $2$ out of $2 \cdot 2$ .

$d = \frac{2 \cdot - 4}{2 (2)}$

Cancel the common factor.

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

Rewrite the expression.

$d = \frac{- 4}{2}$

Cancel the common factor of $- 4$ and $2$ .

Factor $2$ out of $- 4$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$e = 0 - \frac{{(- 8)}^{2}}{4 \cdot 2}$

Simplify the right side.

Simplify each term.

Raise $- 8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 0 - \frac{64}{8}$

Divide $64$ by $8$ .

$e = 0 - 1 \cdot 8$

Multiply $- 1$ by $8$ .

$e = 0 - 8$

Subtract $8$ from $0$ .

$e = - 8$

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

$2 {(x - 2)}^{2} - 8 = y - 14$

Step 5

Move all terms not containing $x$ to the right side of the equation.

Add $8$ to both sides of the equation.

$2 {(x - 2)}^{2} = y - 14 + 8$

Add $- 14$ and $8$ .

$2 {(x - 2)}^{2} = y - 6$

Step 6

Divide each term in $2 {(x - 2)}^{2} = y - 6$ by $2$ and simplify.

Divide each term in $2 {(x - 2)}^{2} = y - 6$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide ${(x - 2)}^{2}$ by $1$ .

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

Simplify the right side.

Divide $- 6$ by $2$ .

${(x - 2)}^{2} = \frac{y}{2} - 3$

Step 7

Factor.

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $- 3$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Multiply $- 3$ by $2$ .

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

Reorder terms.

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

Find the Standard Form of the Parabola f(x)=2x^2-8x+14

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