Find the Standard Form of the Parabola f(x)=-x^2-2x+24

$f (x) = - x^{2} - 2 x + 24$

Step 1

Write $f (x) = - x^{2} - 2 x + 24$ as an equation.

$y = - x^{2} - 2 x + 24$

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.

$- x^{2} - 2 x + 24 = y$

Step 3

Subtract $24$ from both sides of the equation.

$- x^{2} - 2 x = y - 24$

Step 4

Complete the square.

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

$a = - 1$

$b = - 2$

$c = 0 = y - 24$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 24$

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

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

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

Simplify the right side.

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

Factor $2$ out of $- 2$ .

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

Move the negative one from the denominator of $\frac{- 1}{- 1}$ .

$d = - 1 \cdot - 1$

Rewrite $- 1 \cdot - 1$ as $- - 1$ .

$d = - - 1$

Multiply $- 1$ 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 $e = c - \frac{b^{2}}{4 a}$ .

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $- 1$ .

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

Divide $4$ by $- 4$ .

$e = 0 - - 1$

Multiply $- 1$ by $- 1$ .

$e = 0 + 1$

Add $0$ and $1$ .

$e = 1$

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

$- {(x + 1)}^{2} + 1 = y - 24$

Step 5

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

Subtract $1$ from both sides of the equation.

$- {(x + 1)}^{2} = y - 24 - 1$

Subtract $1$ from $- 24$ .

$- {(x + 1)}^{2} = y - 25$

Step 6

Divide each term in $- {(x + 1)}^{2} = y - 25$ by $- 1$ and simplify.

Divide each term in $- {(x + 1)}^{2} = y - 25$ by $- 1$ .

$\frac{- {(x + 1)}^{2}}{- 1} = \frac{y}{- 1} + \frac{- 25}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{{(x + 1)}^{2}}{1} = \frac{y}{- 1} + \frac{- 25}{- 1}$

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

${(x + 1)}^{2} = \frac{y}{- 1} + \frac{- 25}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{y}{- 1}$ .

${(x + 1)}^{2} = - 1 \cdot y + \frac{- 25}{- 1}$

Rewrite $- 1 \cdot y$ as $- y$ .

${(x + 1)}^{2} = - y + \frac{- 25}{- 1}$

Divide $- 25$ by $- 1$ .

${(x + 1)}^{2} = - y + 25$

Step 7

Factor.

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

${(x + 1)}^{2} = - (y) + 25$

Rewrite $25$ as $- 1 (- 25)$ .

${(x + 1)}^{2} = - (y) - 1 (- 25)$

Factor $- 1$ out of $- (y) - 1 (- 25)$ .

${(x + 1)}^{2} = - (y - 25)$

Find the Standard Form of the Parabola f(x)=-x^2-2x+24

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