Find the Vertex Form f(x)=-x^2-7x-6

$f (x) = - x^{2} - 7 x - 6$

Step 1

Write $f (x) = - x^{2} - 7 x - 6$ as an equation.

$y = - x^{2} - 7 x - 6$

Step 2

Complete the square for $- x^{2} - 7 x - 6$ .

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

$a = - 1$

$b = - 7$

$c = - 6$

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

$d = \frac{- 7}{2 \cdot - 1}$

Simplify the right side.

Multiply $2$ by $- 1$ .

$d = \frac{- 7}{- 2}$

Dividing two negative values results in a positive value.

$d = \frac{7}{2}$

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

$e = - 6 - \frac{{(- 7)}^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

Raise $- 7$ to the power of $2$ .

$e = - 6 - \frac{49}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = - 6 - \frac{49}{- 4}$

Move the negative in front of the fraction.

$e = - 6 - - \frac{49}{4}$

Multiply $- - \frac{49}{4}$ .

Multiply $- 1$ by $- 1$ .

$e = - 6 + 1 (\frac{49}{4})$

Multiply $\frac{49}{4}$ by $1$ .

$e = - 6 + \frac{49}{4}$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = - 6 \cdot \frac{4}{4} + \frac{49}{4}$

Combine $- 6$ and $\frac{4}{4}$ .

$e = \frac{- 6 \cdot 4}{4} + \frac{49}{4}$

Combine the numerators over the common denominator.

$e = \frac{- 6 \cdot 4 + 49}{4}$

Simplify the numerator.

Multiply $- 6$ by $4$ .

$e = \frac{- 24 + 49}{4}$

Add $- 24$ and $49$ .

$e = \frac{25}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + \frac{7}{2})}^{2} + \frac{25}{4}$ .

$- {(x + \frac{7}{2})}^{2} + \frac{25}{4}$

Step 3

Set $y$ equal to the new right side.

$y = - {(x + \frac{7}{2})}^{2} + \frac{25}{4}$

Find the Vertex Form f(x)=-x^2-7x-6

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