Find the Standard Form of the Parabola y=-4x^2-72x-315

$y = - 4 x^{2} - 72 x - 315$

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

$- 4 x^{2} - 72 x - 315 = y$

Add $315$ to both sides of the equation.

$- 4 x^{2} - 72 x = y + 315$

Complete the square.

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

$a = - 4$

$b = - 72$

$c = 0 = y + 315$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y + 315$

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

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

$d = \frac{- 72}{2 \cdot - 4}$

Simplify the right side.

Cancel the common factor of $- 72$ and $2$ .

Factor $2$ out of $- 72$ .

$d = \frac{2 \cdot - 36}{2 \cdot - 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 4$ .

$d = \frac{2 \cdot - 36}{2 (- 4)}$

Cancel the common factor.

$d = \frac{2 \cdot - 36}{2 \cdot - 4}$

Rewrite the expression.

$d = \frac{- 36}{- 4}$

Cancel the common factor of $- 36$ and $- 4$ .

Factor $- 4$ out of $- 36$ .

$d = \frac{- 4 (9)}{- 4}$

Cancel the common factors.

Factor $- 4$ out of $- 4$ .

$d = \frac{- 4 \cdot 9}{- 4 \cdot 1}$

Cancel the common factor.

$d = \frac{- 4 \cdot 9}{- 4 \cdot 1}$

Rewrite the expression.

$d = \frac{9}{1}$

Divide $9$ by $1$ .

$d = 9$

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

$e = 0 - \frac{{(- 72)}^{2}}{4 \cdot - 4}$

Simplify the right side.

Simplify each term.

Raise $- 72$ to the power of $2$ .

$e = 0 - \frac{5184}{4 \cdot - 4}$

Multiply $4$ by $- 4$ .

$e = 0 - \frac{5184}{- 16}$

Divide $5184$ by $- 16$ .

$e = 0 - - 324$

Multiply $- 1$ by $- 324$ .

$e = 0 + 324$

Add $0$ and $324$ .

$e = 324$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(x + 9)}^{2} + 324$ .

$- 4 {(x + 9)}^{2} + 324 = y + 315$

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

Subtract $324$ from both sides of the equation.

$- 4 {(x + 9)}^{2} = y + 315 - 324$

Subtract $324$ from $315$ .

$- 4 {(x + 9)}^{2} = y - 9$

Divide each term in $- 4 {(x + 9)}^{2} = y - 9$ by $- 4$ and simplify.

Divide each term in $- 4 {(x + 9)}^{2} = y - 9$ by $- 4$ .

$\frac{- 4 {(x + 9)}^{2}}{- 4} = \frac{y}{- 4} + \frac{- 9}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 {(x + 9)}^{2}}{- 4} = \frac{y}{- 4} + \frac{- 9}{- 4}$

Divide ${(x + 9)}^{2}$ by $1$ .

${(x + 9)}^{2} = \frac{y}{- 4} + \frac{- 9}{- 4}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

${(x + 9)}^{2} = - \frac{y}{4} + \frac{- 9}{- 4}$

Dividing two negative values results in a positive value.

${(x + 9)}^{2} = - \frac{y}{4} + \frac{9}{4}$

Factor.

Combine the numerators over the common denominator.

${(x + 9)}^{2} = \frac{- y + 9}{4}$

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

${(x + 9)}^{2} = \frac{- (y) + 9}{4}$

Rewrite $9$ as $- 1 (- 9)$ .

${(x + 9)}^{2} = \frac{- (y) - 1 (- 9)}{4}$

Factor $- 1$ out of $- (y) - 1 (- 9)$ .

${(x + 9)}^{2} = \frac{- (y - 9)}{4}$

Move the negative in front of the fraction.

${(x + 9)}^{2} = - \frac{y - 9}{4}$

Reorder terms.

${(x + 9)}^{2} = - (\frac{1}{4} (y - 9))$

Remove parentheses.

${(x + 9)}^{2} = - \frac{1}{4} (y - 9)$

Find the Standard Form of the Parabola y=-4x^2-72x-315

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