Find the Vertex g(r)=r^2-6r-55

$g (r) = r^{2} - 6 r - 55$

Step 1

Rewrite the equation in terms of $x$ and $y$ .

$y = x^{2} - 6 x - 55$

Step 2

Rewrite the equation in vertex form.

Complete the square for $x^{2} - 6 x - 55$ .

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

$a = 1$

$b = - 6$

$c = - 55$

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

$d = \frac{- 6}{2 \cdot 1}$

Cancel the common factor of $- 6$ and $2$ .

Factor $2$ out of $- 6$ .

$d = \frac{2 \cdot - 3}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 3}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 3}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 3}{1}$

Divide $- 3$ by $1$ .

$d = - 3$

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

$e = - 55 - \frac{{(- 6)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 6$ to the power of $2$ .

$e = - 55 - \frac{36}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = - 55 - \frac{36}{4}$

Divide $36$ by $4$ .

$e = - 55 - 1 \cdot 9$

Multiply $- 1$ by $9$ .

$e = - 55 - 9$

Subtract $9$ from $- 55$ .

$e = - 64$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 3)}^{2} - 64$ .

${(x - 3)}^{2} - 64$

Set $y$ equal to the new right side.

$y = {(x - 3)}^{2} - 64$

Step 3

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = 1$

$h = 3$

$k = - 64$

Step 4

Find the vertex $(h, k)$ .

$(3, - 64)$

Step 5

Find the Vertex g(r)=r^2-6r-55

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