Find the Standard Form of the Parabola y=-1/2x^2-x+1/2

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

Step 1

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

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

Step 2

Subtract $\frac{1}{2}$ from both sides of the equation.

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

Step 3

Complete the square.

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

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

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

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

$b = - 1$

$c = 0 = y - \frac{1}{2}$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - \frac{1}{2}$

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

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

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

Simplify the right side.

Dividing two negative values results in a positive value.

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

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

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

Divide $2$ by $2$ .

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

Divide $1$ by $1$ .

$d = 1$

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

$e = 0 - \frac{{(- 1)}^{2}}{4 (- \frac{1}{2})}$

Simplify the right side.

Simplify each term.

Raise $- 1$ to the power of $2$ .

$e = 0 - \frac{1}{4 (- \frac{1}{2})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = 0 - \frac{1}{- 4 (\frac{1}{2})}$

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

$e = 0 - \frac{1}{\frac{- 4}{2}}$

Divide $- 4$ by $2$ .

$e = 0 - \frac{1}{- 2}$

Move the negative in front of the fraction.

$e = 0 - - \frac{1}{2}$

Multiply $- - \frac{1}{2}$ .

Multiply $- 1$ by $- 1$ .

$e = 0 + 1 (\frac{1}{2})$

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

$e = 0 + \frac{1}{2}$

Add $0$ and $\frac{1}{2}$ .

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

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

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

Step 4

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

Subtract $\frac{1}{2}$ from both sides of the equation.

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $- 1$ .

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

Divide $- 2$ by $2$ .

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

Step 5

Multiply each term in $- \frac{1}{2} \cdot {(x + 1)}^{2} = y - 1$ by $- 2$ to eliminate the fractions.

Multiply each term in $- \frac{1}{2} \cdot {(x + 1)}^{2} = y - 1$ by $- 2$ .

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

Simplify the left side.

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

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{{(x + 1)}^{2}}{2}$ into the numerator.

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

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply.

Multiply $- 1$ by $- 1$ .

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

Multiply ${(x + 1)}^{2}$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move $- 2$ to the left of $y$ .

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

Multiply $- 1$ by $- 2$ .

${(x + 1)}^{2} = - 2 y + 2$

Step 6

Factor.

Factor $2$ out of $- 2 y + 2$ .

Factor $2$ out of $- 2 y$ .

${(x + 1)}^{2} = 2 (- y) + 2$

Factor $2$ out of $2$ .

${(x + 1)}^{2} = 2 (- y) + 2 (1)$

Factor $2$ out of $2 (- y) + 2 (1)$ .

${(x + 1)}^{2} = 2 (- y + 1)$

Factor $- 1$ out of $- y$ .

${(x + 1)}^{2} = 2 (- (y) + 1)$

Rewrite $1$ as $- 1 (- 1)$ .

${(x + 1)}^{2} = 2 (- (y) - 1 (- 1))$

Factor $- 1$ out of $- (y) - 1 (- 1)$ .

${(x + 1)}^{2} = 2 (- (y - 1))$

Multiply $- 1$ by $2$ .

${(x + 1)}^{2} = - 2 (y - 1)$

Find the Standard Form of the Parabola y=-1/2x^2-x+1/2

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